As long as the key is legit and matches the edition you want to install, it will work.
Also: On the websites where I can "aquire" a disc, there are downloads that seem to have like every single version of vista in the same download. I dont know how this works, so can someone help me?
With Vista and Windows 7, all the different editions of the install discs are made from pretty much the same master. Every disc has the necessary files to install any version from Starter to Ultimate (which makes the "Anytime Upgrade" feature easier because all you have to do is buy a key). The only difference is a very small (1 or 2 kilobyte) configuration file that tells the installer what version to install. If that file isn't present, the installer will kick up a window that asks you which edition you want to install. Just select Home Premium on that screen and your Home Premium key will work.
Before you wait for the install disc to download, does your mom's PC have a recovery partition? (Some OEMs like Compaq, HP, and I think Gateway have an emergency partition you can boot from that acts like a set of recovery discs. It saves them the expense of manufacturing DVDs for every model.) If it does, use that instead because you know it will work and you don't have to wait for 4 gigs of DVD to download.