How can I figure out where this water is coming from?
You have asked a lot of questions at once here.  Let's start by saying that it is very unlikely that you would be able to determine the exact location of the leak from the top, unless it is a simple problem with the sealing around the shower mixing valve handle escusson trim or an obviously loose drain flange. If the problem is in fact under the floor, then you will have to remove part of the ceiling to gain access to the drain pipes for inspection and repair. It is not uncommon for joint to crack or come undone. I wouldn't consider removing the shower floor until it is inspected from below.  Depending on how comfortable you are digging into a semi difficult project like this will determine if you want to hire a pro or tackle it yourself. It will most likely involve removing a section of the ceiling, repairing some PVC pipe or making a new joint with pipe and couplers, primer, pvc cement etc., then repairing the hole in your ceiling.