Hey mate. All pets in the game do the same damage, play what you think is cool and suits your hunter.
That being said, each pet specialization has its own abilities. A wolf, being a Ferocity pet, will have Primal Rage (Blood Lust) whereas Hati, being a Tenacity pet, will have the Survival of the Fittest defensive cooldown.
This is a nice guide on what pets have what abilities, that you may find interesting.
Edit:
I didn’t quite answer all of your questions, but here are some pets I use for each type of content (I don’t do arena):
Mythic+
I’ll use a Ferocity pet that has Primal Rage if no other classes in the comp have Bloodlust.
I’ll use a Spirit Beast otherwise, for the heal and defensive
Random Bg’s
I’ll usually use a Spirit Beast for the heal and defensive
Sometimes I’ll use a Cunning pet for getting out of snares, or a Chimaera for slows
Raids
Same reasoning as Mythic+
Solo play
Spirit Beasts generally, if things don’t hit too hard
For Visions and harder hitting mobs I’ll use a clefthoof for their extra armor and healing received
Basically every pet deals the same damage if you exclude their spec/special abilities.
In addition to the website perrin shared, https://www.wow-petopia.com/abilities.php this also is a very good website to find the pet you want. It also shows where you can tame them.
For your other questions, I’m not a BM hunter so I defer these to the others
The ferocity leech is far better than the spirit beast heal and core hounds are more tanky due to their damage reduction skill. The leech applies to your pet as well so it’s even better as your own personal tank and has crazy self healing in AOE situations. Core hounds also deal more damage when they are being melee attacked because they have a reflect type ability. Spirit Beast hardly offers you anything for soloing in comparison to Core Hounds, just a heal that you’ll hardly ever bother using.
Anything for normal/heroic/mythic dungeons, doesn’t really matter. I go for ferocity Chimaeras for the leech (which means the healer doesn’t have to help you as much) and the extra AOE damage ability.
I always use ferocity in Mythic+/Raids (unless it’s been predetermined by the leader that I should use something else). Specifically Chimaeras because they have an extra AOE damage ability. Spirit Beast can be useful in Mythic+ but the heal is pretty minor in the grand scheme of things. It’s rare that it’ll significantly help the group. As with other instanced dungeons, better to have the leech so the healer doesnt have to help you as much and can focus on others.
Devilsaurs for PvP because they reduce the healing the target receives by 25% automatically and that debuff has 100% uptime. It is essential for PvP and every hunter should be using it. Riverbeasts/Direhorns/Hydras/Lizards can be used instead if you want a Tenacity pet for PvP for the extra damage reduction cooldown, they also have the healing received debuff. I prefer having the Leech from Ferocity though. Scorpids/Carrion Birds/Ravagers/Wasps can be used as a ferocity one as well, they also have the healing received debuff. It doesn’t have to be Devilsaur.
Summary of what I think every BM Hunter should have in their toolkit:
You’ve received great answers already in this post so I can’t add much else to what they’ve already said because it’d just be repetition but, for context, pets used to vary in damage and wolves were considered the best for PvE content.
Blizzard standardised pet damage across the board to allow flexibility / customisation with the focus being on special abilities rather than damage output.
This means you can choose to have whatever you want but some pets are more ideal in certain situations but it won’t be a deal-breaker if you differ. Happy hunting!