If you're planning to get a UPS for this, make sure it does NOT have an auto cut off if there is no load. Most of the UPSes have this feature and connecting a modem to a dedicated UPS will shut it off after some minutes as it detects no load. I've tried this myself on an APC and a D-Link Digipower UPS, both switches off automatically after a while.
weird. i have never seen a UPS shutting itself off because of low load. and just today i noticed my APC UPS was off when I woke up. only modem and router were running on it as the PC was shutdown. i first thought that it ran out of battery as there was no light in the house. asked mom and she said that the power had just gone for around 15-20 minutes. UPS did not start when i tried to start it. it only turned on when the power came back. not sure if it went down because of low load or the APC junk is malfunctioning again.
Yes, it is possible, If you can connect through (USB to modem pin), you will have access to your router.
I think all routers having same kind of power pins, So prepare a wire with one end should be usb pin and the other end should be router pin(buy the router pin at store) then plug the usb pin to usb ports of your laptop or system and other end pin should be connected to your router. Simpleeeeeee and Effective......
yes, very much possible.
my routers are rigged with that (only tp-link td-w9870 is not yet rigged, because i am working on the bootloader, so i can enable, ubifs and allow boot straight off USB)
used the 4+4 cell phone 3.7v batteries, connected them in such a way that a power switch lets me select 5v/9v/12v and then via a barrel plug i connect that to my devices.
pretty much doable.