No
Let's take this feature at a time...
Quick build
This is just flatly wrong. This class is a half-caster, they should not prioritize their casting stat. Glancing down below, this class would always want to prioritize Dex--they are restricted to Light Armor no access to the Mage Armor spell and they don't have Cantrips for steady damage output. So they'll want a high Dex for AC, and to make use of Ranged or Finesse weapons--Str is not a good primary for a class that is restricted to Light Armor.
Proficiencies
Light armor only? This pigeonholes this class into a Dex-build if they want anything resembling durability. They are not a dedicated caster--don't even have Cantrips.
Draconic bond.
Basically a Familiar that can attack at this point. However, unlike a Chain Warlock's familiar, the Pseudodragon does not consume one of your attacks in order to act--as-written, this is a complete secondary character.
Spellcasting
You have the spell range of a half-caster, but weirdly scaling efficiency. Without playtesting, it would be hard to determine how this pans out.
That said, your spell list is basically a grab-bag of the most useful spells in the game drawn from every class's list. Balanced fairly well by the fact that they are mostly (all?) Concentration spells
As an aside--you need to rename this feature. Classes that don't follow the standard rules for spellcasting call their magic feature something else--like a Warlock's Pact Magic.
Fighting Style
You get the same full range of fighting style options as a Fighter does. There is basically precedent for this with a Paladin.
ASI
Normal, whatever, moving on.
Draconic Growth
And here is where the balance starts its nose-dive. Assuming you take a Blue or Gold dragon, you just added a CR 3 companion. At a rough ballpark based on experience, a CR 3 monster is roughly the equivalent of a 4th to 5th level character in combat. So, at 5th level, you are basically controlling 2 characters of roughly similar combat effectiveness.
For comparison, the wolf of a 5th level Beast Master ranger has 1 less AC, 1/3 the HP, and does a fair bit less damage while attacking. And, most importantly A ranger must give up an attack of their own to let their wolf attack. And that's not including the dragon's breath weapon.
I would also note that about this point--your 'pet' is lining up to be as good at Persuasion as your party's social lead unless you have someone with Expertise in Social Skills
Improved Bond
Seems handy in a pinch...not a huge deal, moving on.
Draconic Tendencies
Again, not a huge deal...but I would note there's nothing in there about what to do if you already have the Survival and Nature skills
Draconic Growth
So, the wyrmlings usefulness has gradually been tapering off and keeping it alive has gotten hard. That's gone again.
Assuming you went for optimization and took a Gold Dragon...you are now a level 10 player with a CR 10 companion. Again, based on experience, I ballpark that a CR 10 monster is at least on-par with a level 14 PC. Now you have a pet dropping three (very powerful) attacks per round, as well as their breath weapon--and almost certainly has more hp than your party's Barbarian.
Extra attack
So...now your character is up to 5 effective attacks per round? 2 from themselves, 3 from their Dragon.
Riding Master
A fragment of the Mounted Combatant feat...moving on
Greater Bond
I can see this stacking in pretty terrifying ways. It will always work unless aimed at a creature with Evasion.
Increased Draconic Tendencies
Fluff, moving on
Draconic Growth
Ah-hahahahahahahahahahahahahahahaha
Why are you still in a party? You have a CR 17 companion (significantly more powerful than a level 20 PC) that has Legendary Actions and Legendary Resistance. Other classes get a 4th Attack or regenerating class resources--but you? You get a pet legendary monster.
Subclasses
Weapon Master
Level 3
Seems niche and kinda weird..whatever
Level 7
Eldritch Knights don't get this until level 18
Level 18
Stronger than the Mounted Combatant feat, and that's some pretty serious damage...
Spell Master
Level 3
Okay, now you're almost a half-caster. Still really inefficient
Level 7
The lack of a range on this is questionable--otherwise this doesn't seem too out of place
Level 18
Again, pretty potent, but not nearly as broken as your pet flying apocalypse.
Summary
You have a boss monster as a free-acting companion. There is no possible way this is balanced against the core classes unless the storyline gives them some way to acquire creatures of similar power. Which you could do, of course--but that's not what you asked.
Ultimately, the character themselves is quite lackluster--with underpowered spellcasting, no extra attack until level 11, and being stuck in light armor.
However, they have a boss monster as a pet. The way gameplay with this class would go is that it'd be kinda meh until level 5--then you'd be the strongest character in the party by far. That would then taper off as enemies caught up with your dragon, though you're still controlling 2 full-fledged independent characters--til you had another massive surge in power at level 10 that would largely carry through until max level--with another (frankly obscene) power surge at level 20.
Almost
The class is fairly well written and shows signs of editing. Most of the problems left with it are stylistic textual errors any experienced player or GM will have no trouble correcting and the authors probably didn't notice existed (e.g "You can cast animate dead" but it needs to say "... and it is a Summoner spell for you", Major Magic should recharge when you take a long rest, not the minion, it should say you can only have one minion at a time, etc). The remaining serious issues I see are:
Forging is broken. You don't get proficiency with Smith's tools (or any other artisan's tools) and the class has no tool proficiency and that's dumb. More pertinently, the subclass gives you as many or more traits and abilities as any other subclass at every level.
The class is only balanced for 4d6 drop one stat play. Otherwise the minion's base stats are too good.
The minion has too many hit points. It's fine at low levels, but at high levels it will be a serious problem, especially since it gets them all back more or less every fight.
Blood Transfer can function as a ritual cure spell, which otherwise does not exist in 5e. This out-of-combat healing is beyond anything else seen in the system. This isn't necessarily a problem, since it helps the minion not outshine other frontliners in the party since they also heal to max between fights now, but that's still a problem for balance
Rapid Summon is a serious problem. You can summon a new minion with every action instead of taking 10 minutes. That means unless your minion dies in one round, you can restore it to full health if you want.
I'm guessing the minion moves as you direct with no action cost to you? That could definitely use stating.
You get extra ASIs. Those are for rogues and fighters, MAD classes with no inherent spellcasting. Why does this SAD class, which has inherent spells, get them? It shouldn't, and should instead have the normal 4/8/12/16/19 ASIs.
I think it should be fairly easy to solve these problems, though. The most important change I would make would be to make creating the minions expensive, so that players are incentivized to not reforge the minion into a completely new being after every fight. I would probably give each house a different cost, to help with the flavor of the minions being created in different ways:
Necromancers: 1 humanoid corpse per hit die. You can use beast corpses instead but it takes an hour instead of 10 minutes, and you can't use Rapid Summoning to lessen the time if you use Beast corpses.
Marionette: 10 lb/ hit die total of wood, clay, wax, and cheap dyes
Forge: 10 gp/ hit die of rare metals
Madness: two or more live beasts or monstrosities whose HD add to exactly the desired total. You can use dead beasts or monstrosities instead but the ritual takes 2d4 hours instead of 10 minutes.
The costs themselves are spitballed and should not be taken as serious suggestions. The point is that, by adding a non-trivial cost though not necessarily in gold, you limit the ability of summoners to use their summon as a freely replenishable bag of hp. They still can summon a new creature with 10 minutes of work and thus be awesome at summoning, they just have a reason to not do that all the time and to keep their minion alive and resummon it instead of a new one. This also fixes Rapid Summoning because the 10-minute timer is no longer the supposed balancing feature against creating a new minion.
Best Answer
'Balance' covers more than just 'balance', if you see what I mean
It's a good thing to ask whether your creation balanced, as in whether it is more or less powerful than existing options.
But there's also the question of balance as in 'is a good addition?', 'is it out of place?' 'is it useful?', 'could it even have been published in 5e in the first place?' and wider questions. I'll address the latter first.
Why are you adding it?
If the answer is as simple as "because it's fun to try and make stuff", fair enough, by all means homebrew it, and if you're with a bunch of mates rolling around casually, then often no-one's going to mind. Homebrewing new content can be enormous fun for the brewer. I make new classes all the time.
However, for a more serious campaign where balance is being considered carefully, as it sounds you are, then a lot of fellow players [or your own players, if you're DM], might want more justification than that. And to do that, you need to answer 'yes' to the following questions:
I've found most players and DMs are as concerned with 'why' than the nuances of power balance. The more detailed and difficult questions you need to think about for the latter are as follows:
(Please note I discuss these aspects of balance checking generally, using hypothetical examples and those from my own homebrew, before discussing the Alchemist specifically each time.)
"Does my class step on another class's (story) toes?"
This depends on how specialist you want classes to be. Are you aware of the 3.5 'Basic' variant where there were 3 classes? (Warrior, Expert, and Spell caster?). The 5E basic rules divide spellcaster into Arcane and Divine. The 5E full rules split arcane into two, divine in two, and warrior into three (fighter, barbarian and monk) before adding magic.
In my high-magic, high-tech, gods-at-war setting, I have added two new divine classes, a Favoured (Soul) and an Invoker (or 'Siphon', TBC). These are just making new specialisms of 'Divine Caster'. The Favoured is a servant of a deity, just as a Cleric is. Oh! I'm stepping on story toes. Or am I?
In my setting, Clerics are trained, organised servants of the pantheon, whereas Favoured characters are untrained, devoted to one master above all. So am I interfering with the Cleric's role? Certainly some PCs and NPCs who would have been Clerics might now be Favoureds. I'm stealing some of the Cleric's player base, perhaps. Is that a problem?
If your class is stealing a player-base (even just for story purposes), it is something to be aware of and that many could be uncomfortable with. But then, introducing Bard, Sorceror and Warlock into the basic rules is stealing the Wizard's players. The Nature Cleric and Ranger steal Druid players. Where do you draw the line?
I would draw the line here:
Does your Alchemist step on another class's (story) toes?
Are you stepping on any story toes with an Alchemist? I would say no. Casting a spell (examples: fireball, teleport) is not the same as alchemy (which goes wider than just potions - we could be talking self-augmentations, poisons, skin grafts, who knows?).
When people say "Just play a Wizard and call it alchemy"...
Opening asker, if you think, like me, that a character should be able to say "I'm a Dwarf Alchemist, behold my concoction!" instead of "I'm a Dwarf Wizard and this Fireball is actually me throwing a bomb really far", then your Alchemist isn't stepping on story toes.
"Does my class step on another class's (mechanical) toes?"
Ah, now we have to express our class's unique place in our story with a unique place in the mechanics. Probably: but on the condition you're not performing another class's mechanical role better, you may not have to be that unique to still have an acceptable class.
Let's take a setting which is relatively peaceful... why the heck should all clergy automatically have armour and weapons? Let's make a 'Priest' or 'Adept' class. Take the Cleric, strip out all armour and weapon abilities (including in the domains, maybe adding Divine Strike to cantrips), and give them Divine Recovery.
Yes, a lot of the time we're playing one of a Cleric's mechanical roles (buffer/healer), but then so do the Bard and Druid. Do we have a mechanical difference which reflects their story role?
Giving them Divine Recovery means their key mechanical difference is ingrained in the story difference. A Cleric is great at being in battle. Our Priest/Adept needs rest - needs peace and quiet. As the whole point is that we're making a peace-time, no-trained-for-battle sort of divine agent, the mechanic fits the story perfectly, even if it's not that unique.
Even when stepping on toes a bit, if your mechanic is at least a bit different, and mechanically expresses a story difference, then it's probably alright.
"Does your Alchemist step on another class's (mechanical) toes?"
Hmm. Are we stepping on the Wizard's mechanical toes? It depends on what mechanic you use.
As you're using spell slots, you need to ask yourself how that mechanic is actually reflecting a story difference between alchemy and wizardry. Personally my thought would be to only duplicate low level spells (2nd attacks, 3rd buffs, or something), but have it almost at will. Why, if I can make and carry one fireball-potion today, can't I make one a day for a week and then go nova with 7 when the fight comes? If I'm going adventuring, why can't I have a backpack full or kit?
Perhaps the alchemical components spoil, AND require the brewer's own blood, or something along those lines. I'm toying with an alchemist myself, and one possible idea is to make the proficiency or constitution bonus worth of potions etc. on a short rest, which last 8 hours before spoiling. (This is in addition to duplicating cantrips as long as they have a standard toolkit on them).
Is it too powerful? Is it too weak, or partially too weak?
Honestly that all depends on what you do with duplicating spells. If you're duplicating up to Fireball (which would be really nice), maybe you should only be able to get that at-will at level 18-20, because that's when a Wizard gets level 3 spells at-will. However, level 1 and 2 spells, the Warlock gets at earlier levels.
Your cantrip system up to 4d6 is less damage than other classes. Even if he can whip out literally any elemental type, bear in mind other get 4-6 cantrips so could have most elemental types available whenever they want. Just make it elegant. (going from 1d4 to multiple d6s is a bit odd: keep the 4 or 6 consistent)
It is hard to give you feedback without seeing more - but then it's probably hard for you to make more without getting feedback, I know.
Note, I've compared mostly to the Wizard, but if you brew stuff on a short rest or something, you will need to compare to the Warlock or Cleric Domains too.
As an exercise to try out balance, put your class in a hypothetical story situation and compare her to how a Wizard/Warlock/Trickery Cleric or whoever would do.
Maybe you can't come up with a specific plan, but thinking over those scenarios should help you decide what your class should and shouldn't be able to do, so you can think "Actually, if I give her delayed blast fireball, that just makes me a Wizard with Con instead of Wis, so maybe I'll make all her spectacular effects require preparation and slow application"
I hope that gives you some ideas to think about for balancing homebrew classes generally and some specific feedback on your Alchemist ideas.