I just put an Amsec BF6636 in a spare room in my basement. Right now it is sitting on the carpet and pad with a concrete subfloor. I intend to bolt this sucker down.

What kind of anchors should I use? Is it ok on the carpet or should I pull it up? If I do end up pulling the carpet then I'd probably lay down some vinyl stick on flooring or something similar, would that be adequate for a moisture barrier?

I do not know how thick the concrete is and I am worried that drilling all the way through would be bad. Maybe seal the holes with some sort of silcone caulk afterward?

Please advise.

I'd leave the carpet

Drilling all the way through is ok. If you had water under the slab it would probably be coming up around the perimeter of the slab already. Silicone might not do anything but it wouldn't hurt
The concrete should be around 4" thick, a 3" anchor will work fine. I used these: Red-Head 3/8" x 3" Sleeve Anchors

If you're around Lexington I can give you some

My favorite for concrete is concrete screws like the DeWalt screwbolts. Don't know what brands you have. Concrete screws are easy to install and easy to remove completely.

Use hilti kwik bolts, you can get some concrete anchor epoxy if you are concerned about sealing it.

I got the safe bolted down yesterday using some redhead wedge anchors. When I initially  placed the safe in the room it was leaning forward quite a bit and the door would swing open pretty quickly under its own weight, so when I was bolting it down I tightened the two anchors in the back first to try and get it leveled out. The two in the back are pretty much as tight as I can get them and the safe is still leaning forward a bit. Not as bad as it was when unbolted, but the door will still open on its own out to 90 degrees.

Can I put some shims or something under the front of the safe to help get the door to be neutral or should I just get a door stopper and live with it?

Yes, shims are a normal way to get a safe level.