Q; Can I just cap these off and put them in headlight bucket? Or will this change the draw (pull) of current enough to mess with my new halogen bullets.
Just cap them and forget them. You will never notice the current change.
2nd Q. If I find some strictly running lights that will work, can I hook them up to these striped wires with a ground and have separate running lights?
Yes and No. Yes, hooking up the running lights as you described will work with this exception: The front running light on the right side turns OFF when the right turn signal is on, and the same with the left. If you want running lights with both lights staying on all the time, you will need to find a different circuit.
The advantage with the stock setup is that the turn signal flash is more pronounced with the dual element bulb (bright/off vs. bright/dim). With separate turn signals and running lights, the running lights will be a little quirky.