Is there a simple tool for fixing invalid geometry in edit mode? I find myself often in a situation where I build an object out of simpler forms, well always ;) However when putting these together I sometimes have invalid geometry I must fix.

I wonder whether there is a tool for fixing this in edit mode?

The "Boolean" modifier operates on objects. I however want to operate on a small part of a complex object.
The knife tool is a step in the right direction. With this there is the problem I must hit the intersections precisely myself, which is not possible.

An example:
Before – 5 sticks intersecting the long main stick


After – Geometry corrected. No more intersecting faces


Is there a way Blender can do this of its own (selecting multiple faces and Blender splits the edges and faces as needed)?
There is an addon called BoolTools that can help with this.  

It's a two step process.  First select the entire mesh and separate by loose parts, then union the component parts.