Do I run 12-2 wire to a wall thermostat and heater off a 20 amp breaker? Does the black and white wire go to a double pole breaker?
It is safe to assume you mean 500 watt and not 500 amp.
Yes, you should run 12/2 cable from a double-pole 20A breaker to the thermostat and then from there to the heater. This is the easiest way to run this circuit. Be sure to re-mark the white wire to a "hot" color with tape or easier yet a permanent marker. Use a color like red, blue or black.
Be sure to use a double-pole line-voltage thermostat.
500 amps is a pretty decent baseboard heater.
A 500 watt heater doesn't need 12/2 wire. 14/2 is fine. It's only pulling 2 amps. Yes, the black wire goes to one side of the 15 amp double-pole breaker and the white wire goes to the other side.
The separate thermostat will probably be a control rather than a power switch, this means small signal wire can be used instead if 14/2. Check the instructions, control voltage is a factor.