can I run an Ethernet cable out of the back and then out to the outhouse into (ideally) a powered switch
Yes but you would normally run a patch cable into a socket, solid conduited/armoured cable from there to a socket in your outhouse, then plug the switch into that with another patch cable.
It's hard to terminate exterior grade cable into an RJ45 plug, much easier to punch it down onto a socket. You want external cable as it has better UV and moisture resistance properties, eg black and full of gel.
Yes but you would normally run a patch cable into a socket, solid conduited/armoured cable from there to a socket in your outhouse, then plug the switch into that with another patch cable.