Hrmm... you really need to figure out how to get into the router config, that's probably where the problem lies. Once you do you could maybe enable NAT for kicks, that may speed up your latency (which may be contributing to your timeouts). And check all the settings in the modem, PPPoE and all that stuff.
*EDIT
And who's DHCP-ing you? The router or the modem? Try setting it to the modem. I had some config problems with mine when I had the router giving me DHCP, but it was nothing like this (I couldn't ping the modem at all).
*EDIT2
Aah, just thought of something. Try running through the router without changing any settings, downloading something, and running a constant ping to the modem. If you can ping the modem the whole time even when you lose your download, the problem is probably bad PPPoE settings in the router.