My vote would go to Highmountain Tauren guardian druid - prancing around in moose form (the chillest travel form by far), unkillable in bear form. It’s got heals, movement speed, resilience and being HMT just makes everything calm.
You pretty much just named all the classes in the game. For real though friend it is all in what you find to be the best at solo content, which is what you seem to be looking for. Warlocks, Hunters, Death Knights, Demon Hunters, etc. all are good for what you mentioned.
The big question is, is what can you enjoy playing that you wont get burnt out on quickly?
Then feign?
But i like both 
Seems redundant to use both. Both drop combat, right?
There are times feign is down and shadowmeld comes in handy its just a QOL issue and shadowmeld doesnt have a time limit.
So Night Elf Hunter would be nice choice but again rep is needed for expansion (extra gear)
From my experience, playing any ranged class is quite chilled out. Melee’s can get frantic.
So for 100% relaxing and chill is a BM hunter what pets should i go after (boosted) ?
I feel pretty relaxed playing my Arcane Mage. This mage has been with me for a
long time.
Demon Hunter u can glide ur way out of these terrible terrain.
Probably BM hunters for the pets or Guardian Druids cause you can pull multiple mobs with Moonfire and not have your HP go under 50%.
Frost mage.
Hunter for sure.