Yes, you can, but it has the same requirements you do. If a part only shows up from breaking a part, then the palico can only have it if you've broken that part. It's literally just additional rewards for each successful plunderblade attack.

I guess you can pick pretty much everything. I just got lucky and received a nerg gem with the plunderblade.

It's more like an additional loot reward at the end of the mission than a carve. Your palico has a chance to get anything you have a chance to get, meaning for example if you've broken both horns off of Black Diablos, it could have a Black Twisted Horn+, but if you haven't then there's no chance it will. Since you always have a low percent chance of getting a gem, that means it always has a low percent chance to also get a gem.

Rather, sometimes it's slinger ammo, sometimes it's a material, it seems to be random. This is from the Palarang ability.

No, you don't get less materials, you get the same amount you would have even if you didn't have this. What it does is 2 things: 1) Sometimes makes the monster drop materials on the ground like when you knock it down or it fights in a turf war, though sometimes this ends up being a pod instead of materials, just keep an eye out for when your palico says to pick it up. 2) Your palico will have monster materials on the 2nd row of its own loot screen at the end of a hunt. So, you're not seeing more or less materials on your own loot screen, but you will see extra stuff on theirs.

theres a mod for that, i forgot the full name but in the mhw world mod tab on nexus mods just type in pillar and you should find it. it makes materials shine a lil white beam up to find them easier

Look for a mod on nexusmods called lightpillars, makes all item drops easily noticeable