Yes, on Sprint and Verizon, lower signal levels means it will burn through more battery as it has to send more power to the radio to maintain a connection.
Nothing you can do. When I had Sprint, if I were stuck in meetings all day, my battery would be dead before I went home - because there was very little service in the conference room (1-2 bars).
It all depends on the coverage each provider has in each area. In some places, AT&T will have a better signal while, in others, Verizon or Sprint will.
Get an AT&T Account or buy a car charger. May improve on its own though with Sprint's Network Modernization Project. One of the goals of the project is improved coverage, which should mean stronger signal.
This is exactly what I've found. In our office, there's maybe 1 tiny bar or "no service". My phone will be dead by the end of the work day, but if I turn off cell data in settings (forcing the phone to use WiFi instead), the battery will barely drop all day.