Yeah when the phone has low signal (1 bar out of 5), the phone tries extra hard to obtain a signal, thus killing the battery, and also making the phone warm.
Not at all. Low signal is caused by poor reception or cell tower issues. This results in your phone to use more battery as it attempts to maintain the connection whilst searching for a stronger one.
If you're in a area where you know you won't have signal for a while - such as underground on a train - switch to Airplane Mode. You will save a lot of battery as the phone won't constantly be making failed attempts at finding a signal.