1) There is no Windows 7 edition called "student and home". MS did offer a cheap student upgrade version a few months ago but that may no longer be available. The Home premium edition should work perfectly well and in fact is designed for the home segment. You can get a comparitive overview of various editions on the link below. Windows 7: Compare Windows 7 Home Premium, Professional, and Ultimate
Whichever edition you pick, you'll need to choose between an upgrade disk and a retail disk. An upgrade disk upgrades your existing license to a Windows 7 license, so your current os license cannot legally be used again. A retail disk is a full-fledged new license and will cost more than the upgrade one. You can check out the price difference at Amazon. In functionality, there would be no difference.
You can also buy a cheap system builder OEM copy from Newegg. But that copy will be tied to the PC it is first installed on. The license is not transferable.
2) I presume by C: partition, you mean the one on which vista is installed. If you have any data files on it too (docs, music, video, pics), best to backup to an external media before switching to Windows 7. In any case, always try and keep the data files on a partition other than the OS partition.
If you choose to do a clean install of Windows 7, your C: drive will be wiped clean and Windows 7 installed. If you choose to do an upgrade install (dont confuse this with the upgrade disk), vista will be upgraded to Windows 7 and your installed apps and data files should be fine. You will be presented both the options during the install. But remember, upgrade installs can be buggy and cause problems later. IMO, best to do a clean install. You'll need to install all your apps and tweaks again in that case.