That at heat lamp looks small. You might need a bigger one.
A CHE is like a bulb and screws in the same socket.
I would recommend a wider/bigger dome and a 60 watt CHE thermostat controlled.
Probes measure temps for a thermostat. They never go in the substrate.
please get a temp gun.
UTH goes on hot side. Lamp/dome goes in the middle of the tank or a little closer to the hot side, depending. You will have to experiment.
Also. What is securing your top? You definitely want screen top latches to secure. If not, all of this is for naught. Your snake WILL escape.