I would definitely use the computer out of its case. I am currently building a bartop and built it with the plan of keeping my computer in its case, and I could have saved a lot of space by not doing that. It's not that it'll be overly huge with the case, but as was said, smaller is better with a bartop.
My bartop is 18" wide and 30" or so deep (guesses off the top of my head. Your best bet is to grab the monitor you're gonna use, and rough up an idea for space needed for computer parts and everything else. I actually put a piece of MDF right up to my monitor to get a rough idea, sketched something out, then went from there.
As for horizontal/vertical... you really have to consider what you're gonna play. I'm going horizontal because most of the games I play often are horizontal. Vetical games aren't terrible on a 15" or 17" screen, but it does get small quick. If you think you're gonna be playing mostly verticals, just go vertical to avoid the even smaller screen.
I'm using a 450mhz Athlon and have had good results with a .55 set so far. I haven't played much, but it seems to run most games fairly well. You'll be even better off of course.