As far as I know the closing of all applications does only concern the installation of a program. I'm not sure how important it is though, because I usually neglect the advice and install a program with everything else left running. I think it might be more of a disclaimer. A download itself can be done simultaneously with other running programs and processes. When you view multiple websites at the same time those pages have to be downloaded temporarily to your pc as well in order for you to see them, and that goes fine simultaneously as well. The only thing that may happen is that it slows down your pc. Or if you download multiple files at once, your internet connection may lose its speed.
I tend to keep downloaded files on my pc until eternity, just because you never know when you need them again (you should see my room, same policy there

). I do however rename them as soon as I download them, because the names they're given originally aren't usually the clearest ones and you may not be able to remember what a file's for after some time has passed.