if I understand correctly, just about any PC you find can handle this. the video card would be the only issue here. you need to determine what kind of monitors you will have connected, and what kind of inputs do they have available. if you are using only PC Monitors, then make sure your video card has dual outputs (either VGA or DVI...you can use adaptors if you need to) if you are planning on connecting this to a standard or hi-def TV, then you will need a card with a video output (composite, s-video, or component) some Hi Def displays have VGA or DVI inputs, but that would really depend on the model.
you can get remote controls that connect to your PC to control it like a standard av component. as for software, you have some options..it really depends on what you want to accomplish. if you go with windows media center edition, it will have alot of the funtionality that you are looking for already built in. or, you could probably use a front end like MALA with some heavy customization to launch your other applications.
personally, if I were doing this, I would choose XP over Vista, it is much more stable at this point in time, and doesnt require as many system resources.
I am assuming that since you have all of these questions, you probably aren't comfortable with building your own PC. (if I am wrong, i apologize) So, when selecting a pc, stay away from celeron processors, i'd personally recommend either a Intel Core 2 Duo, or an AMD Athlon 64X2 processor, at least 1gb(1024MB) of ram, and if you are planning on storing alot of movies/music on this PC, i'd go with around a 500gb hard drive. the only other thing to make sure of is the video card. anything listed with "integrated" or "on-board" graphics is bad. you want a dedicated video card with multiple outputs. but like i said before, you need to know what types of inputs are available on your monitors so you can get a card that is capable of connecting to them.
i don't know what your budget is like, you could build a nice PC for around $700. if you choose to go with one of those $300 dells, make sure you don't skimp on the RAM. at that price range, they will probably all have integrated graphics, just make sure they have a PCI express 16 slot, so that you could install a better video card.