I love, LOVE this book and have read it many times. I wonder many of the same things you do. My thoughts...
First, is this Bruce Wayne somehow immortal? Or is he the original's grandson or something?
It's never explained, and I've searched for any interviews where Paul Pope explains this and turned up nothing. I agree with Bryan that the symbolism is Batman as an immortal concept. That said, I do think this is Bruce who has somehow made himself immortal. I can think of dozens of ways he could have pulled this off (the Lazaraus Pit being one obvious one). The DC Universe is a weird and wacky place!
Second, what happened to the government?
It went completely fascist and became super corrupt and oppressive. The visual of the White House being under guard and fences is one of the more striking images in the book.
Third, am I correct in thinking that some of the Arkham inmates were turned into government pawns after the place was scrubbed?
There's no indication of this. Based on what Gordon says, none of them should be around at all. That said, I wouldn't be surprised if we found out some escaped before the purge or did become government pawns.
That leads me to my great wish that Pope would do a follow up series. I love the story and his art style is so distinct every page is a pleasure to behold.