Does GPS signal quality influence battery life? Does bad/weak GPS signal affect battery life? For instance, when in the city downtown with lots of tall buildings or in the woods or mountains, will running GPS drain the battery faster?
No. Bad cellphone coverage decreases battery life because the phone has to transmit with more power to be able to communicate back to the cell tower. It's like when you can barely hear someone hollering at you: you shout louder too to make sure they can hear you. But GPS is a one-way signal: the phone only receives it, it doesn't transmit anything. It doesn't matter how much attenuation there is: either the phone can hear or it can't.

The only way a weak GPS signal would cause more power use is if you leave the screen turned on waiting for an app to get a position fix :-)