How do I safely pick up a rabbit, and can you pick it up by its ears?

Definitely not by the ears. The old rabbit-in-the-hat trick is not recommended. Basically, the safest way is if you just kind of lean over the rabbit back to front and then pick them up from underneath. Be sure to support their shoulders and bums carefully, and then they're facing outwards so that if they get upset and kick, they're not kicking against anything. It's safer for their back because sometimes, if you have them turned around and they kick against you, they're just so strong that they can injure their back that way.