Find and Refluff a Class or Ability
The inflict line looks like a handy set if you just want 'damaging blood'. The Dread Necromancer (3.5) class gets an ability called Charnel Touch that does small amounts of negative energy damage, why not.
Pick something that you can fluff similarly, preferably with a spell list (spell lists are great, pick the stuff you can refluff as blood-related, take eschew materials, ignore the rest).
Play a Black Blade Magus and fluff the black blade as being made out of your blood.
Write A Class
It's really not hard to write a class. Keep in mind the basics (level x 2 = overall attack bonus, level x 5 = overall damage per round, level x 10 hp for melee, half those numbers for spellcasters, 3/4 those numbers for half-casters), and just invent some thematic abilities in the style you want. You want to be a melee blood-user, right? So this class should a) give you a magical weapon that scales with level (blood slash, or whatever), b) give you some blood-based means of defense that scales with level (blood armour? fast healing?), c) some wildcard ability(s) (blood teleport? sanguine fury? rip blood out of someone and save vs death? burning blood defense?) at appropriate levels d) something unique and out of combat, like blood healing or blood scrying or assimilating people's memories by drinking their blood or something.
Call it the Cursed Blood Scion and you're cookin' with gas.
Cursed Blood Scion
BAB: Good
HD: D10
Fort: Good
Ref: Poor
Will: Poor
Skills: 4+Int
Cursed Blood Scion
1 Cursed Blood, Bloody Frenzy
2 Tainted Blood, Fearsome Bloody Body
3 Murderous Blood, Blood Armour
4 Ancient Prophecy, Taste Taint
5 Know Blood, Terrifying Mien
Cursed Blood - Your blood can be used as a weapon. Striking someone with your blood, or a weapon formed from your blood, does 1d8 + half your class level as an enhancement bonus + 1.5 your str. Attacking someone with your blood is an attack action, and you add half your class level as an enhancement bonus to that roll.
Bloody Frenzy - If a creature dies or is disabled by lethal damage within 5' of you, you can cover yourself in their blood and go into a frenzy. You immediately may strike out at another foe within range with your Cursed Blood attack. This ability triggers even if it isn't your turn and you didn't kill or disable the creature. The creature may be an ally or an enemy.
Tainted Blood - Half the damage from your Cursed Blood ability is now Negative Energy damage. In addition, creatures must make a fortitude save to avoid being Shaken for the duration of the encounter when struck by your Cursed Blood ability. Once they make this save, they do not have to make another for 24 hours.
Fearsome Bloody Body - Whenever you use your Bloody Frenzy ability, you may immediately make an intimidate check with a +5 bonus vs any enemy with a clear line of sight. Whenever you succeed on an intimidate check by 10 or more, that foe is Frightened instead of Shaken.
Murderous Blood - When loosed from your body, your blood does it's best to kill people. Whenever you suffer a blow that deals hp damage, your opponent takes an amount of damage equal to half your class level, half slashing, half acid. If you should be knocked unconscious or killed, your blood forms into a small ooze-like creature and attacks enemies to the best of it's ability (see: Minor Blood Ooze creature entry). Should you be revived, it immediately returns to your body.
Blood Armour - You can form your blood into a protective armour that covers your entire body. This is a Full-Round Action, and counts as Full Plate Armour with an enhancement bonus equal to half your class level. It also looks fantastically badass, and gives you a +2 morale bonus to intimidate checks. You are fatigued as normal for heavy armour if you wear your Blood Armour while running, traveling, or sleeping.
Ancient Prophecy - An ancient prophecy sleeps within your veins, doom hidden in the curse that writhes within your blood. This allows you to tell the future by how cursed and doomed you feel. Once per day, you may use Divination with a 90% success chance but only to determine the likelihood of bloodshed in any specific circumstance.
Taste Taint - If you taste someone's blood, you can tell if they are Evil, Vile, Cursed (as if by Bestow Cursed), a Black Blood Oracle, or otherwise 'cursed' or 'tainted'. No spell can disguise this, but all you can tell is they are 'tainted' or not - not which specific source of 'taint' it might be. Tasting spilled blood in combat is a move action if you are the one who spilled it - standard otherwise.
Know Blood - If someone whose blood you have tasted is within your class level x 5' of you, you know which square they are in, and which person it is. You never forget the taste of someone's blood.
Terrifying Mien - If you are covered in someone else's blood, you can make a Demoralize intimidate check as a move action, instead of a standard. If you instead perform it was a full round action, you can attempt to Demoralize every enemy within 30'.
Walk The Red Road(8th, paimon blade dance), The Blood Remembers(long distance travel through blood), Sanguine Frenzy(murder things), Call of Blood(make it hard for foes to retreat), Bloody Allure(intimidate as diplomacy), That Red, Red Wine(better living through blood ingestion), Cleanse and Purify(disease removal), Bloody Grin (intimidate as will save, 12th), Blood Shield, Birthed In Carnage, Hemophage, Swim the Red River, Smell Blood, Evil for Evil, Blood for Blood, Death is a Metaphor, Blood-Drenched Heritage, Your Blood is My Blood, Crimson Vampire Knight, Flow In A Million Veins, Walk A Thousand Battlefields
There is nothing in the 5e rules (at least that I'm aware of) that either allows or prevents the acquisition of a non-combat non-magical pet.
That means that whether or not you can have a pet is completely and totally up to your DM.
As to whether or not there is a balance issue related to having a pet, I would not be very worried about it. It may occasionally provide a very small benefit (I'd maybe let it provide advantage once a day if I wanted something explicitly mechanical for it), but in general it's not going to be important and is merely a flavor element for a character.
That said, if you did want a mechanical element to a pet, the Find Familiar spell would be a good place to look to get a non-combat effective pet that does have some mechanical utility.
In summary, there's nothing in the rules that allows or prevents this, so consult your DM, but do so knowing that it's probably not going to be a mechanical concern (especially if it's spelled out explicitly that it can't be). If there is a need for more mechanical weight, then find familiar is a good model.
On the more general aspect, I can only speak to 4e, but it had a theme called Animal Master that gave you an animal companion that was basically just a vanity pet with a small mechanical benefit (you could use it to gain combat advantage once per day). It was a minion, so not very combat effective and was mostly a flavor piece with a minor benefit.
Best Answer
Answer for D&D 5th Edition
Answer 1 to Question 1
You'll have to work with your DM, and perhaps consider a Warforged character.
There is no Player Character Class in the Players Handbook with the description of animated armor you called out, but the Warforged seem similar. During your discussion with your DM, explore how a Warforged character might fit your idea. This race is for the Ebberon setting, and is ported in from earlier editions of D&D (3.5e). Now that the Eberron: Rising From the Last War is published, it may fit better.
This gives you a point of departure in your discussion with your DM, and could apply to any class in the Players Handbook. Keeping it a secret might not be possible, if you take that option.
Answer 2 to Question 1
Since you may be referring to a construct from the Monster Manual, Animated Object, (p. 19-20 of MM; p. 9 Basic Rules, DM) then you need to proceed as follows:
Discuss this in detail with your DM, since you wish to create a monster as a playable character. In all editions of D&D this can be fun, but it brings with it some risks of not fitting with the rest of your party, and balance issues run amok. In the DMG Chapter 9, the Dungeon Master's Workshop, some tools are available for you both to work with.
In order to "make it stronger" (I presume by that you mean gain capability as you progress in level) apply the same XP goals as other characters for levels 1 through 20, then agree on the special skills and traits that accrue as levels progress: you'll want to fit in special abilities at levels 2, 3, 5, 6, 10, 14, 18, and maybe 20. (Or, drop one of the later ones, that may be a bit much ...)
By level six, your homebrew character should look something like the stats of "Animated Armor," which has six hit dice.
Conclusion: Stealth checks at disadvantage, if even allowed. Good luck keeping this a secret.
Challenges
These strengths and weaknesses need to be addressed before you make it into a playable race.
Working with your DM is critical for two reasons:
Answer to Question 2
This answer depends on how you and your DM answer Question 1. Your decision on a Warforged character, or a playable suit of animated armor (monster/construct) as a PC will result in a homebrew character. Any homebrew (which can be great fun!) must (I repeat must) be compatible with:
This takes us back to answer 1:
Work with your DM to see how you can fit this idea into his campaign.
Warning: if anyone succeeds in giving your character a heart, you may turn into a flesh and blood human being! Be wary of picking up the wrong magical axe, young girls with dogs, lions, and scarecrows. :-)