Cut path at intersections
So I have the following
and I would like to remove the lines inside of the circle. I've tried the scissors tool, joining them then using the scissors, and outlining them using the pathfinder window then using scissors.
I think you are trying to make it too complicated. It really can be very simple.
You could use the Add Anchor Point tool to add two anchor points to the straight lines, then use the Direct Section tool to delete the parts of the line and any remaining anchors you want.