Unless you mean to greatly exaggerate the immediate kick, and then automatically recover back to a value close to what we have now, which might work.
Yes, that's it exactly.
I guess what I'm trying to propose is to make actions that would be autonomic in real life, automatic in game, if that makes sense. For example, if you would have an instinctive, reflexive reaction to something IRL (something your body does without a conscious command), then your avatar should also do that in game.
Also, regarding weapon movement - I'm not contradicting myself - just explaining badly the first time

Sorry, I"m really bad at explaining my ideas - they make perfect sense in my head, and I find it really hard to explain things from a point of view of someone who doesn't just know what i'm talking about.. (I really suck at teaching people things, for this reason).
Okay, here goes a better explanation of my cover idea:
1. If you are within a few feet of a coverable object then you will automatically go into cover mode if not already
2. you will not be able to peek/aim out of cover until suppression effect wears off, however you can blind fire. Bear in mind this only occurs in strong suppression, which means bullets are landing or passing very close to you and would likely hit you if you did stick your head out.
3. I also would say that you would ONLY 'cling' to impenetrable cover - to make sure you dont end up getting stuck to a wooden fence and then get shot through it. If it's not possible for it to distinguish between penetrable/impenetrable cover, then scrap the whole idea.
The idea behind this is simulating a fear response - instinctively jerking your head/body back faster than you'd be able to do it consciously