Detective Career: If you still have left over evidence but think the last case is over, you can still analyze it and still be able to put clues on the board even though you haven't been to a new crime scene yet. Some of it fails and some of it does give new clues.
Seven clues are best to narrow down who the real suspect is on any case. Five is too general. Also, if you are unsure if two look a lot alike. Approach them with friendly interactions, click Get To Know, which will tell you the trait. And narrow it down even more, especially if one of your clues is a trait (which is learned most of the time by having more than five clues).
ETA: It's much better to wait and ignore the put out an APB until you have six or seven clues instead of just five.