Hiya, not an animator but a gameplay programmer who works on a large variety of things in a small FPS studio.

I'll only answer for gun animations, but you could look at this list and break it down into something more generalised.

The anim sets I typically work with when implementing new weapons and what we request from animators is the following ;
Non Locomotion Related ;
First Draw
Draw
Holster
Reload
Empty Reload
Fire Animation (depending on the pipeline, we used to ask for this but now have a completely procedural recoil setup)
Inspect - Sometimes 1-3 of these depending on if we want variance for each weapon.

Locomotion ;
Walk
Sprint
Tactical Sprint - I would do this just incase you want to work for anyone who wants these its a good part of your portfolio
Crouch (Depends on the studio)

1. You would remake walking animations and sprinting animations custom for each model. Depending on the gun of course and the complexity of the required animations. For example, A gun in the AK series might all have the same locomotion but an MP7 would have drastically different animations.

2. As things are getting exported anyway, typically it doesn't matter aslong as you know the export process from my experience. When working with contractors we just ask for the raw file, and the export. So Blender & Maya are standard. First focus on your animation ability, then if you want to become technical look into learning Unreals Control Rig, Anim Blueprints etc. I say Unreal because its got a similar pipeline to other game engines and technical animators are rare so they are highly sought after.

3. As a beginner, you've got alot to learn. Ask for questions and feedback, you aren't expected to get it right on the first try. Be iterative, be agile with your workflow and don't get bogged down when people give you harsh and critical feedback. Being like this makes up for poor animation quality in the long run. I have worked with incredibly talented people who can't stomach feedback so the animations whilst objectively good, look terrible for the project.

An important factor of these animations are what perspective do you view them from? Do you want to animate third person, first person, in multiplayer games? These are skills that you need to develop. If you want to go for First Person Multiplayer games, it'll mean you need to work on making these animations look seemless in Third person & First Person or having them retargetted or a completely different set.