I'm completely new to this site and I've already submitted a run, but I've done another run afterwards which was faster. Can I submit the faster time or am I stuck with the original? If it is possible, can I just edit my original, already submitted run or do I need to submit a completely different one?

It's better if you submit the new one. Your previous submission will become an obsolete run and will be hidden in your profile if anyone wants to view it.

[quote=Liv]Can either edit it, [...][/quote] please never edit your runs to submit a new one. it messes with all sorts of statistics and invalidates them.

^ I know someone who does that with every one of his runs, including a game I've also done. It disappoints me greatly (honestly disinclines me a little from doing that run).