[codesuidae]

You can power your relay with a circuit like this:

                       VCC7.2v
                        +
                        |
                     o--o
                     |  |
             snubber V  C relay
              diode  -  C coil
                     |  C
                     o--o
                        |
                 ___  |/
 From ZZSE >--o-|___|-| ne2222
   2.4v       |  2k   |>
              |  ___    |
              o-|___|---o
                10k     |
                       ===
                       GND

This way the relay coil will be energized using your higher voltage circuit. The 10k resistor can probably be omitted. The transistor is any N-channel small signal transistor, Radio Shack has them.

A motor will use the most power when it is stalled, so they tend to get hot under load. Running unloaded they only get hot if you are exceeding their rated voltage by quite a bit. The armature, the spinning part inside with the wires wound around it, is what gets hot. The current flows through the wires and heats them up.

Failure modes vary, usually it will just get hot enough to melt the wires somewhere (often near connections since they have higher resistance). Sometimes high temperatures can melt solder connections or cause moving parts to bind or seize.

Good heat sinking on the motor case helps, but heat can only leave the armature so fast, so you can still burn up the motor at high power.