Thats a good question. First find out what your IP is, and say it is x.x.x.20, then your gateway should be x.x.x.1, and your subnet should be 255.255.255.1. I am am not totally sure of the subnet. You may want to go direct to the modem, reboot and check with a IPconfig, but DONT use the IP you see there. Use what ever IP you get when going to what to http://www.whatismyip.com/ with the router hooked up. But please note, setting up your router this way, by it's very nature, is only temporary. If you leave it set up this way, eventually your internet will break.
About 1 and 3, the thing to remember when bypassing equipment to isolate a problem; absence of proof is not proof of absence. There can be situations where you power cycling a modem fixes a router problem, which I know may sound counter intuitive. Or where power cycling a router can fix a problem, when really it was the modem that creating the problem. Thats why on top of simple trial and error isolation, you need to walk through the problem logically. My question to myself was "What problem with a network occurs every 12-36hours, that is fixed with a power cycle, but doesn't effect the modem's signal lights." First thing that comes to mind is a IP issue. There is nothing to "fix" when it comes to a modem and a IP. Yes, there can be issues where the modem will give out the wrong IP, but those are broke/not broke types of issues, not intermittent. But I think I am over explaining myself. 