Edit: I do not play this game anymore so I am not up-to-date on which weapon appears on which of the new map. Or whether these achievements can be obtained by playing the TDM mode. If you leave a comment with any feedback I'll gladly update the solution though.
There's not much to this, but here are the Valid weapons:
-AWM (only found in carepackages)
-Winchester 94 (Not found on Erangel)
Knockdowns do not count, however headshotting a downed enemy that you yourself knocked down, does count.
As for most of the [kill x amount of players with x weapon or equipment] achievements, you can down an enemy by any means, so long as you finish the player off with whatever you're trying to progress the achievement of
Edit: Anyone who downvotes, please be so kind as to tell me why so I can improve the solution accordingly.
-VSS Worked for me, however multiple people have said it doesn't. Don't count on it working.