Cpl. Person is an Immortal.And probably only recently had his first death, which is why he seems to have no concern over getting shot at outside Al Muwaffiquya; he's not thinking it through and realizing it's still possible, even if unlikely, he could lose his head, or that he really doesn't want to deal with the repercussions if so many people were to see him get up from being shot. Colbert knows (and probably saw him die for the first time) and lets him keep a sword hidden under all the crap in back of the Humvee. Person's comment about how awesome it would be if Colbert "rode into battle with a sword" is an in-joke they share, possibly even a way to preemptively explain Person's sword as just another part of his usual sillyness, in case anyone ever sees it.
Rudy is from the Otherworld.Which is why all the guys want him. Besides, he's not pale enough for a Meyerpire.
Sgt. Colbert is from a cyberpunk future.Weary of the weird clothing and camp ways of the future, he travels back to the 21st century to find his true identity in war.
The Marines used so many military jargons so often because they were trying to teach them to ordinary people.Knowing that Wright will inevitably write a book about the experience, they used as many military jargons as possible in front of him so that he would have to describe them in the book, making the ordinary civilians understand the military better.
Captain America was named as such, because he took the Super-Soldier Serum.Obviously, given his status amongst the Marines, he was given a faulty variant of the Super-Soldier Serum, which caused him to be, well, a little unhinged.