It does have a PCIe x16 slot so you can put a new graphics card in it, however it will be held back by the fact that it is a 1.0 PCIe. You will need to make sure the bios is updated as well.
Beneficial, yes! You are a few generations behind by now, but keep in mind with changing your mobo you will also need to upgrade your CPU, and depending on what connectons your PSU have that as well.
Yeah, an LGA775 socket, since then there has been LGA 1155 Sandy and Ivy, and LGA 1150 which is Haswell, with the Haswell refresh Devil's Cannyon coming out sometime this yer.
sounds like it might be cheaper just to get a whole new computer at this point but I really would love it if I could just do a few things and have it work well