Should I care about topology after applying booleans? Should I care about the topology of an abject to which I have applied a lot of booleans? In my case I am not planning on doing any transformations on the face with the messy topology (see picture below). It's basically finished. Do I or do I not have to redo the topology and why?
If the object isn't going to be animated and the topology doesn't cause any problems elsewhere (e.g. because hard edges are used and bad topology becomes apparent), then you don't have to. 

If you'd like to retopo a static object, but don't like to spend a lot of time on it, I'd suggest taking a look at Instant Meshes an open-source tool for automatic retopo operations.