[MuGen]

I don't know of any programs or internet activity that could possible overload your router causing it to shut down. But I'd say you have some sort of parasite in your computer.

For the initial problem at hand, it could be Spyware/Adware because it hijacks the algorithms in your browser to either log keystrokes or activity so you may want to run a sweep on that.

And if it were a Time out problem Firefox usually gives a 'Connection timed out'