Bicycle and bike are correct. I fact bike is an abbreviation for bicycle !
Of course, the course is still in beta, there are many missing translations, just report.
This sentence makes no sense. If you were getting a bike for free, you would not be renting it, which involves payment. So I translated “pujcit” to borrow, but that was, of course, not accepted.
When I see půjčit I think of borrow. Although usually if I borrow it, it is for free so borrow for free seems weird too.