Is there really no delete or cut? So I made a fin on my fish that I don't like. So in every app existing I select and delete. 

In real sculpture I just cut off or carve away.

How do I do this in Blender?

I tried selecting vertices in edit but it won't mirror so it becomes asymetric. I have used booleans but the remaining surface becomes super wierd when I want to smooth it out to get a fresh start.

Select + delete in sculpt.

Or even better, a carve tool that eats away chunks and not leaving wierd remains.
You can delete selected vertices/edges/faces by pressing Delete or X.
To make the selection symmetrical, use either Mirror selection or a Mirror modifier

Mirror selection across X axis