Do soldiers in a combat zone police their brass?

What about those little clips that the M1 fired that would go bing and fly into the air when they fired their last shot? Did anyone pick them up for reuse?

How about M-16 magazines? Do the guys just rip through them and toss 'em, to not get distracted from the battle, or do they carefully collect them to refill for the next day’s battle?

What about resupply? Does a soldier just go up to a supply tent and ask for a full load of ammo, or is each bullet and magazine dear?

In combat, there is no policing of brass, unless you do not want evidence you were in a certain area. In the subject line I thought you meant polish!

What about those little clips that the M1 fired that would go bing and fly into the air when they fired their last shot? Did anyone pick them up for reuse?

I believe we are still talking about the same type of items, but I will let someone who is an expert on the M1 come along.

How about M-16 magazines? Do the guys just rip through them and toss 'em, to not get distracted from the battle, or do they carefully collect them to refill for the next day’s battle?

It would probably depend on the situation. If in combat you were running place to place and you dropped your mag on accident, you would probably not stop to pick it up. However you are trained to pick it up because there is a good chance you will need to refill it. So, in the best case scenario, yes. But not if it threatens life.

What about resupply? Does a soldier just go up to a supply tent and ask for a full load of ammo, or is each bullet and magazine dear?

Ammo is distributed to the individual unit. In a non-combat situation it is definitely rationed and counted. In a combat situation, it is basically given out to the units, which disperses it down the chain. If no rounds are fired, the ammunition is turned in as long as combat operations are over. Rounds are counted the best as possible, but in combat a lot is overlooked.

In combat, one tends to be a bit more pragmatic. WRT to magazines, generally one would try to hang on to them, to refill later, but of course other matters of attention may take priority. IME any kind of ammunition resupply would, when possible, be in the form of fully loaded magazines (at least, that’s how I would organize it, were I in charge of such things), but Murphy’s law would probably recommend keeping a grip on magazines as much as practical.

My Uncle had a surplus case of M1 ammo. It was preloaded in the 8-round clips, so there would be no reason to collect the expended clips.

You wouldn’t want to discard the magazines from an M16. The ammo comes in 10-round stripper clips packaged in bandoliers. There’s a widget that you attach to the top of the magazine that allows it to be loaded from a stripper clip.