Not sure. How would one get the signal from the modem to the other router. I understand some routers have a WAN port for that.
Yeah, a short length of CAT5.
I'm not sure using two modems would be a good idea. It might be better to get one working well. How far away are the devices from the wifi signal? Can you raise the modem higher? Can you position it to remove obstacles? Is there a firmware upgrade you can get?