Can I hang a bathroom shelf with command strips or something similar?
I would not risk hanging a glass shelf with those strips.  They are meant for hanging things directly from (like a towel, hair dryer, etc.), not a shelf that sticks out.  Instead of just pulling down on the strip, it will pull down and away, almost a guaranteed recipe for broken glass.

Unfortunately there is no way to anchor something to the wall without proper anchors.  If you are able to get the shelf anchored into a stud then the hole should not be all that large - it's the drywall anchors that tend to make larger holes.

If the wall was concrete you might be able to glue it to the wall, but this will leave a larger mess (leftover glue that someone has to scrape off) than screws in drywall.

Perhaps you could offer to repair any damage you are responsible for?  Otherwise you should get a free-standing shelving unit.