I would have to say frost mage/ww monk/ frost dk then for you i would advise making a trail of each and testing them out apart from that nothing else i can say as you are very specific on what you want.
Warrior or death knight. Both of them got some healing, both of them are plate characters so you have more armor. 3th option is monk. You have to try them and see which one is cool for you.
You get gnome hunter, MarkMan ! Do not allow a pet , say LonerWolf and if the group ask you , tell them NO this is bad and the worst and terrible that thehunter get to say they do not need the pet to be the MarkMan ! And then RUIN this game for pet to do the heroicsm spell
WW for better sustain (self-heal is okay-ish but it’s still better than Exhilaration) and tank/heal offspec (you can play FW healer as pseudo-DPS). Good mobility overall - FSK, 1 to 3 rolls (or torpedos), tiger’s lust for speed burst and de-root/snare.
MM for juicy aimed shot slaps with kinda less amount of buttons and more straight-forward rotation. Pet is optional for utility like BL (can’t use Spirit Mend in MM so will not cover that one). Less survivability but still have some (turtle/bear/fd/sotf). Disengage and Cheetah for speed.
Also, maybe Fury Warrior – not many buttons to press, good self-sustain, okay-ish mobility with leap and charges, good animations. For big numbers you can switch to Arms. Also, zug.
Obliterate build frost DK hits every point you’re looking for. Has an easy rotation with not many buttons, has no pets you have to look after, does individual big crits, has decent sustained healing, in TWW rider of the apocalypse also gives decent mobility.
Shadow Priest, mindbender w dark ascension wings build.
Use the glyph to reduce shadow form (it’s much better than it used to be).
Gnome
Low buttons? Can be if you pick the right talents, if you are going to copy paste from a website then I can’t help you.
No pets/summons (tentacles and mindbender do their thing)
Self-sustainable
Big numbers? Sure it has dots but then BOOM BOOM!
Good mobility if you pick the right talents - speedy feather, speedy bubble, speedy cloud of darkness. No I can’t remember the spell names off the top of my head.
Graphical effects? Well it’s purple boogley gorgeousness and the void hero spec in TWW is blue sparkly gorgeousness.