If I am traveling and I need a place to daven, is it OK to use a relatively clean bathroom, being that the toilet flushes away anything that would be an obstacle to praying. Many times it is hard to concentrate in public places, but in a private bathroom it can be quiet and best place to concentrate.
Yes. You are correct. Just make sure that any smells are gone.
Can I listen to shiurim while doing my makeup / hair in the bathroom ?
Assuming you are properly dressed, toilet is clean, no odors etc. then yes.
Oh, and its after you have recited birchas Hatorah as well.