Maximum PC

Until I can afford a custom-built laptop, I was considering buying a mid-range ASUS to tide me over. Is the GeForce GTS 360M good enough to play games at high to ultra-high at 1080p? Or will I have to dial down the graphics settings or resolution in order to get playable frame rates on it. I've never used an Nvidia card before, so any information on the card's capabilities will be greatly appreciated.

What games? Generally you'll be limited to middling settings on new games with a GTS 360M at 1080p.

I was thinking of more graphically intensive games like Fallout 3 and Far Cry 2.

You can run it at 1080p but for intensive game, maybe medium-high for good performance.

Would it be worth getting a 15-inch with that GPU? Or buy a 17-inch with a 5870 and suck it up with the extra weight?

Depends if you go places with it, but a 15-inch gaming laptop is not going to be light either. It's really about the size - a 17-inch gaming laptop can be overwhelming.

I'd be taking it to school with me every day and to my grandparents' house when I go.

no way then. Taking a 15" laptop to school is enough of a chore. I did that for 3 years, and I wish I could have replaced it with my M11x way before I did.
If you're going to take it to school, I think you'd be best off going with an HP Envy 14. Its plenty fast for games at its 1600x900 resolution, and is far more portable than bigger laptops. Its well worth not being able to play at the very highest detail levels.

Have you also considered building a desktop and getting a budget gaming laptop? For $1000, you can build a desktop thats far faster at games than the highest-end $5000 laptops, and then get a cheap $700 gaming laptop just for mobility. Desktops also last longer, because you can't really upgrade laptops.

Is the Envy 14 a quad-core?

I need a quad-core laptop because I'll be using it for a video editing class in school.