All the description needed is in my projects under my profile along with file of project.
Solidworks has no problem with intersections- it can extrude even with their existence. The problem arises when you have gaps in your sketch. Even then you can extrude closed loops in your sketch by choosing Selected Contours manually under Boss-Extrude.To repair your sketch use Repair Sketch command in SolidWorks to find out where your gaps are.
You also may be creating a 0 thickness (like two cubes touching at only the corner) that SW does not like. Try simplifying your sketch in a few different ones and maybe un-check the merge option as this will create a new body and eliminate any 0 thickness.