cantripsdnd-5espells
There are a number of class features that provide benefits to 'spells'. For example, the Eldritch Knight's attacks can give an enemy disadvantage on saving throws from the EK's spells.
Do cantrips count as spells for the purposes of such features?
Obviously, in previous editions when cantrips were just 0-level spells, they would have. However, in 5e there is a level of differentiation between cantrips and spells.
There are a few reasons for a spellcaster to be in melee (most of which you mentioned.) I'll cover them one by one.
One is auras that buff nearby allies. The Paladin wins this one. Honestly, no other class even comes close. However, outside that, the Paladin's primary focus is on buffing his own melee attacks with the [X]-ing Smite spells, which isn't exactly what you're after. Also, as a half-caster, the Paladin is almost certainly not what you're after.
The next is spells that damage nearby enemies. The standout here is the Cleric's Spirit Guardians, which, once cast, damages everything around you for up to 10 minutes without requiring any further actions on your part. Other notable spells include the Paladin's Destructive Wave and the Warlock's Arms of Hadar. There's not too many spells in this category, unfortunately. The Elemental Player's Companion addded some good new ones to look for - Thunderclap is a cantrip that's just perfect for a melee spellcaster, and Earth Tremor is a pretty awesome too.
Reason number three is if you want to be attacked for some reason. The Wizard's Fire Shield is probably the best spell for this, but the Warlock's (or Tiefling's) Hellish Rebuke or the Tempest Cleric's Wrath of the Storm ability work here too.
Next up, spells that have their origin on you. Burning Hands, Cone of Cold, Thunderwave, etc. There's a lot of these, but none of them really stand out, no class is particularly better at them, and most of them don't really require you to be in melee anyway. They're good, and you'll want to have some of them, there's just not much to say about them here.
Finally, melee range attack spells. (And yes, I saved the best till last.) There is a definite winner here: Vampiric Touch. A minute of powerful melee spell attacks that do a fair bit of damage and heal you? Perfect. It's available to a few classes, but the one that gets the most out of it is a Necromancy Wizard, because Grim Harvest will get you even more healing when you kill something with Vampiric Touch. If your DM will allow it, the Death Cleric from the DMG also gets pretty good with this spell, with Touch of Death, Inescapable Destruction, and eventually, Improved Reaper (!!!).
I'd probably recommend the Necromancy Wizard or the Death Cleric, but whatever you choose, you are going to run into one problem: concentration. Most of the good spells I've mentioned require concentration, which means you can only use 1 at a time, and you are liable to lose the spell when you get hit by attacks. Which, as a melee spellcaster, is probably going to be often. Concentration checks aren't particularly difficult, but if you have to make a lot of them you might consider investing in a method of gaining advantage or proficiency on them. The War Caster feat gives advantage, while the Resilient feat, Sorcerer, or Barbarian give proficiency. Of course, there are a lot of ways to give advantage on or boost any roll, so you might not find this a problem.
On Wizard schools specifically:
Abjuration has some great benefits, but the only one that is particularly good for a melee is the Arcane Ward, which effectively soaks one hit per day, plus a little bit more every time you cast an Abjuration spell. Not bad, but not great.
Conjuration has Focused Conjuration, which allows you to ignore damage for concentrating (great!) but only for Conjuration spells, of which there aren't many that suit you.
Divination has nothing that works specifically for your purposes.
Enchantment actually has one ability that's great for this: Hypnotic Gaze. It effectively allows you to lock down a single creature in melee.
Evocation is great for evocations (as you might expect), and most of the damage spells with you as the origin are evocation, so definitely worth a thought.
Illusion has an auto-miss reaction, which is cool, but not much else.
Necromancy has the aforementioned Grim Harvest, which is pretty good, and Inured to Undeath, also pretty good (but only when fighting undead), but its other features are more geared around having an army of undead fighting for you, which is pretty powerful, but not really what you're after.
Transmutation doesn't have all that much for a melee, except that you can get a free proficiency in Constitution saving throws, which will help you maintain concentration.
On Shield Master specifically: for you, it's almost completely junk. It boosts your Dexterity saving throws, and nothing else. I'd really recommend against taking it. Heavily Armoured, on the other hand, will take your AC from a maximum of 17 (assuming you have a Dex modifier of +2), to 18, with no investment in Dex required. With the houserule in place that gives you shield proficiency as well, it pushes your AC up to 20 - at this point it goes from good to downright amazing. AC is quite rare and valuable in 5e, especially if you're used to 3.5 and everyone walking around with AC 30.
I was thinking about this too, so here's my build (added an additional Feat from your houserules):
Lvl 4 Tiefling, Archfey Tome Warlock
This build points to forgive the melee and AC options for a more focused caster. I liked your idea of an hybrid, but seems to me that it really needs to be an hexblade, going in the direction of the other classical builds around. Tiefling adds some interesting features, while allowing to reach 20 CHA on lvl 4, increasing the overall effectiveness and chance to hit of the few spells you can cast.
By Lvl 4 you have:
I chose Archfey over Old One for several reasons:
- Spells: Dissonant Whispers and Tasha's seems kinda redundant to me, especially when you can get Hold Person at lvl 4. Sleep it's an excellent spell vs. low level mass enemies, it has no concentration and grows with spell slots. Faerie Fire can grant advantage to the entire party vs one or more hard targets. Both pacts have Phantasmal Force which is a great disable for low int enemies and fun to use too.
- Pact Powers: Fey Presence is an area disable which fits more the theme than Telepathy granted by Old One. At lvl6 Entropic ward is fine, but Misty Escape is simply awesome.
Invocations: I'd go to PimpMyBlast just because this is your only reliable source of damage in a lot of situations where CC or forced movement are not options - If all CCs fails, if you want to blast that minion in the backrows not worthy a spell, in long fights, ecc.. But if you like, you can forgo one of the blast invocations to get rituals, which can give you a bit more flexibility and a Familiar.
Feats: i'd really like going first at all costs with this build, so i'll take Alert. A well placed Sleep or Hold person the first round is no joke. Otherwise Moderately Armored can give you better ac with a shield, but going melee will require War Caster too to be a viable option.
Ability scores in order of importance: Max Cha, High Dex, High Con, Mid Wis, Low Int, Low Str
Role: cc\secondary caster
Playstyle: Disable high treat enemies as soon as you can. Use forced movement (Blast, Thorn Whip) to generate AO from allies and reposition enemies at your vantage. Minor Illusion to modify battlefield (covers, false gaps, ecc), Vicious Mockery to help ally, Shocking Grasp to disengage, Thaumaturgy\Friends as social tool.
Progression: At lvl5 you can get the infamous Devil's Sight + Darkness combo which can transform you into a walking nightmare of disabling madness, very hard to target. Lots of tactics here, disabling sight with plain movement is a powerful option, plus you attack with advantage; just take care of LOS of your allies. If you want more utility you can always get Rituals. Darkness is 1/day from Tiefling so you can also get a lvl 3 spell. While patron's spells are not that great, at lvl 3 you have 2 strong area disables (Fear and Hypnotic Pattern) and Counterspell, which is really useful if you are the only caster in the party.
Here is Lvl detailed progression:
lvl 1
Tiefling: +2 Cha, +1 Int, Thaumaturgy
Patron Fey: 1-turn Area disable
Spells: 0 - Eldritch Blast, Minor Illusion; 1 - Hex, Sleep
lvl 2
Invocations: Agonizing + Repelling
Spells: 1 - Faerie Fire
lvl 3
Pact of the Tome: Vicious Mockery, Thorn Whip, Shocking Grasp
Spells: 2 - Phantasmal Force
Tiefling: Hellish Rebuke 1/day
lvl 4
Spells: 0 - Friends; 2 - Hold Person
Feats: Alert (from houserule 4th lvl)
Best Answer
Cantrips are spells.
There are certain class features that require a spell, or a spell slot, of level 1 or above (The Life Cleric's Blessed Healer feature for instance) and as such cantrips would not qualify for these. For all other spell casting features, they would qualify.