2. Yeah, heavy weapons are great; but, they come with a cost. Having eye for supply is a must! In my games, the guy with the heavy weapon became the focal point for the squad. The others went out of their way to pitch in, because no one wants their machine gunner to run out of ammo, ever.
2a. Resupply is used to exchange supply for gear or gear for supply, which may not be something you can do in the heat of battle, since it's going in and out of unit stores. You can swap gear with each other whenever your characters could reasonably do so. So, when boomer yells, “Shit, I’m out of ammo!” and you toss him a mag, he marks a gear and you mark off a gear, simple as that.
If it’s a heavy weapon, like the flamethrower, where the ammo is rare, bulky, and heavy, then resupply is legit. If it’s machine gun ammo, where the squad is probably always humping belts, then you just do it. There's an element of discretion here. If it makes sense, go with it. The GM tightens the noose by letting them blow through gear, not by stopping them from doing so. Saying yes is the harder move.
2b. That’s a great question! Gear, like you say, is an abstract quantity of stuff that your character has on them. It’s abstract so that you can establish the thing you need in the fiction, right when you need it, without having to plan it in advance and without having to micro-manage inventory.
If the thing already exists in the fiction, then it is established and you can just use it, as though you spent gear on it or scrounged to find it. So, if the machine pistol in the hands of the fallen enemy soldier is established, then you just pick up the gun and run with it. It’s probably fair, in general, to assume that you’ve got at least one move’s worth of ammo for a weapon you scavenge. It’s not gear, because it’s already established, so you just use it. If there’s a question (in the GM’s mind) as to whether the gun is functional or whether there’s any ammo for it, then like you say – scrounge is the right call.
A better way to handle the situation is to avoid being overly-specific when it comes to equipment lying around. Then, you just scrounge and it's all just gear. If there's a salient thing, mention it. Otherwise, wait for the question, make them scrounge, then answer.
An exhaustive search is definitely a scrounge move. Also, scrounge is used when definite spoils haven’t yet been established or when the quantity/quality of the spoils aren’t established or when you might run into trouble while scavenging.
1. In Apocalypse World, the Gunlugger’s signature weapon is a playbook entitlement. In the Regiment, weapons are stuff and stuff is the GM’s purview, so the continued availability and utility of stuff is the GM’s prerogative. There’s no contract. If it’s good for the story, and it sounds like it is, then sure, throw them a bone. But, I’d be a little concerned when a player gets too attached to their character or their character’s things. This is a game where the GM and players alike should “look through sights” at all people and things in the game world.
A more fundamental thing is this – the mechanics are totally agnostic towards ammunition and ordnance. Bullets are gear and guns shoot bullets, so if you have gear you can shoot your gun. The point of gear is to avoid long, heated gun porn debates that contribute in no way to the story. For some groups, mine included, gun porn is Regiment foreplay. The extent you bring ammo compatibility into play (or not) in your game is completely your call as a gaming group. Compatibility could be ignored entirely, as simple as “us” vs. “them,” you can get right down to the I and IS variants of 7.92 mm Mauser ammo, or anywhere in between. So, what I’m saying here is that the Commando, strictly speaking, could scrounge enemy ammo and just use it, if you’re cool with that. Nothing in the game says you can't or shouldn't do this.
Let’s say you're aren't cool with that. You could mark friendly gear with an “X” and enemy gear with a “\” (or vice versa); same with supply. Check off the type of gear used. Then, when you run out of friendly gear, you’re stuck using captured weapons. That’s a simple way to do “us” vs. “them” compatibility. Grenades are grenades, bandages are bandages, and food is food, so there are plenty of other things you could do with captured gear. That’s as far as I’d go with the Regiment. Sure, use compatibility as color in the fiction beyond that, maybe; but, don’t track bullets. That's a different game.
Another thing to keep in mind is that you can scrounge for friendly supplies behind enemy lines too, if it makes sense (e.g. other friendlies in the area, etc.). There might be supply drops or dumps around. You might poach another platoon’s gear. At some point, there are friendly casualties: fallen comrades are good for gear too, if unfortunate.
As a side thought, in WWII and among poorly-provisioned armies of the world, captured weapons and weapon systems were/are used to fill arms shortages. I suspect this is probably even more the case when the fighting gets really desperate. So, just because Jerry should have a Mauser, doesn’t mean you won’t find a Mosin-Nagant on him instead. If that happens to be what you need, then more power to you.