"Hij denkt dat hij iemand is, maar hij is niemand."

I would say it's a burn!

Yes, but I think it's not used as commonly as in English. Dutch probably loaned this usage from English quite recently, since you also can come across a sentence like: hij is een nobody.

Like Susande said we sometimes use it in Dutch, but I would say phrases like "Hij is een nul" (lit. He's a zero) or "Hij is niks waard" (lit. He's nothing worth) are more common.

Shortly, when you have dat, the verb goes to the end. It's the opposite with maar (and en, want, etc), these don't make the verb move to the end. Sorry for this brief answer, but I hope you understood ^^

You can either include it or exclude it; both work perfectly fine. It's basically just a matter of preference. ^_^

Assuming it's your report, since it is the only one in the system and from the same time you posted, you forgot to type he after that: He thinks that is somebody,...