There's no should. We can do it either way, edit or close. It's a pretty good course of action to edit questions like this, though, or at least not a bad one.
We could just close the question and ask the author to pick a question and post the others separately, and often do. Sometimes however if we're clearly going to go down that route, it speeds up the process to just pick one out for the asker (usually the highest quality of the bunch, i.e. the one most likely to stay open) and edit out the others, and give the asker guidance on where to go from here: post the edited-out questions separately, here's how you can check the edit history and retrieve them, here's some improvements you might want to make to those whilst you're busy doing this.
It saves time and several close and reopen votes (and the asker from experiencing getting a question closed in the first place, which a lot of people don't like), and saves us from possibly clearing up a messclearing up a mess if someone wants to jump the gun and answer all of the questions at once, and possibly thus leaving the asker feeling no need to take action. Also, it gives the asker (and us!) one still-open good question, whether or not the asker decides they want to take action.
If the asker rolls it back, we go ahead with the close process. If all the questions are bad quality and there isn't one to pick out that's likely to stay open, it's not worth doing the edit and the question should just get closed with avenues for improvement suggested.
