How to stop a water leak through concrete?
Several parts of your question can't be answered here, as they deal with costs with contractors.  That said, a broken pipe would usually have water flow down, after all, water seeks its own level.  If a drain pipe is broken in some way that clogs it, the water might come back up the drains.

As to what you can do, that depends on how big of a bite you're willing to take on.  If you want to remove a shower, it can be done.  Tearing up a floor isn't complicated, either.  Putting replacements in is a little more difficult, but there are books and websites and youtube videos for any issue you face.  The local hardware shop should be able to give you an idea of what you need and how it goes together when you get to that point.  Your local code office can also tell you what they want to see as far as doing the job properly to code.

Goes without saying, but this is work that generally will require a building permit and probably an inspection.