In your case, however, you could completely forgo a splitter and simply connect the two rooms together with a coax coupler. 3. goCoax, Actiontec and Motorola are all reputable brands. 4. I have no experience with this model. Ubiquiti Unifi and TP-Link EAP APs are very popular on this subreddit.
MoCA will work fine assuming the coax is unused. You probably have all your coax in the house going to one splitter. You'll want to disconnect the rooms with MoCA adapters and connect them via coupler.
https://www.amazon.com/RiteAV-4330136515-Coax-Coupler-Premium/dp/B000V1PAVM
MoCA isn't cheap though. If you've already experimented running ethernet I'd try to make that option work, maybe with a different higher quality cable.
probably. you probably just need to extend the coax cable going into your modem. one of these
https://www.amazon.com/RiteAV-4330136515-Coax-Coupler-Premium/dp/B000V1PAVM
and however long coax cable you need would probably do it.
or maybe you need more ethernet slack, where you could get a coupling for that and do it. just take a look and i'll bet you can figure it out