I’ll say this now, unless you push very high Keys/mythic raids they are all perfectly viable.
Now to answer you:
As a matter of fact BM is miles ahead of everything else in Pve, it’s not that great for pvp tho.
Survival is the best pvp spec, it has more utility and damage then the other two. Still decent for Pve but worse then BM.
Marksmanship is sadly in not that great of a spec right now, it has less damage then BM in pve and less mobility. It’s decent against some comps in pvp but suffers from low cc and low reliable damage. Very good in random Bgs if you can outrange the enemies.
If you want to push high key/arena my advice is to go Bm for Pve and survival for Pvp, I’ll warn you tho you’ll have to find a good weapon for both and have two sets of azerite, it’s a big pain IMO but doable.
As for gear from Pvp: if you open the dungeon finder page, under the pvp tab, you’ll find at the top of the window a bar that if you fill will give you a quest that if completed will then give you the piece of gear.
The Item level of said piece grows with your arena rating if i remember correctly.
To fill the bar you’ll need to get “conquest” this is either acquired from the daily assaults in zandalar/kul’tiras or by doing Pvp (remember that non rated Pvp will give less/no conquest after you get it once, resets daily)
If you have more questions feel free to ask, I’ll answer them if i can :D.