I have a website of my own and a forum on it where I have posted some of the typical things I find and the solutions...many are wireless related but you may want to read through some of them.
http://www.cyberallies.com//support/nfphpbb/index.php
some other sites of use to me over the years are....
http://practicallynetworked.com/
and
http://www.speedguide.net/
let me know what you find....
for it to lose connection so often through the router and not when it is directly connected is suspicious to me....unless your isp is no longer seeing the MAC address of your network card or something which can be "Spoofed" in your router settings...in ther words to make it work when it is directly connected to the computer then entering the mac address of your network card into the router so the modem thinks it is still connected to the internal ethernet card....this should or is not normally a problem with my company anyway but may be what is happenening in some way...just something to try fooling around with...instead of rebooting let it connected drectly to you computer via an ethernet cable.....get connectivity....then bring up a dos prompt and run
ipconfig /all
to get the mac address of your ethernet card...then hook it back up to the router without rebooting the modem and entering the mac address of the ethernet card into the router....it should still think it is connected to the modem directly.