You have 2 seperate problems.
The dome light gets power all of the time. Remove the bulb and check for 12 volts at both sides of the light socket. One side/contact should have 12 volts. If not trace the wiring back to the fuse box. You need the 12 volts.
If you have the 12 volts, then there is an open between the other side of the dome light and its ground. There are 3 places for the circuit to find ground. The light switch & either door switch. I doubt that all 3 went bad, you probably have a broken wire.
The heater problem is probably a wire that you crushed/pinched and it is grounded out. I'm not sure of the routing, but you need to follow the wires from the blower motor back to the switch or relay and fuse box.
When you say the heater was too hot to touch, do you mean the heater core???Blower motor??/