Working through your switch issues first to get rid of the hot light. I will use your switch picture and help work backwards. The schematics list the wire color, gauge and circuit number. It will help for you to look at the schematic when reading this. This is from a ‘68 Polara manual, but basic circuit should be really close.
Clockwise -
Top horizontal - ACC (Accessories), 12 gauge Black - should only feed accessories when key is on. This wire goes to a junction feeding power to the accessory side of the fuse box (12g Black), turn signal flasher (18g black), and windsheild wiper switch (16g pink) Does your heater blower, wiper or flashers work with the key off?
Clock wise vertical ST start - 18g Yellow - Feeds T at the bulkhead junction box turning to grey on engine bay side. and goes to starting relay (shouldn't be causing issues) should only be live while cranking.
3 O 'clock Horizontal IGN2 - 14 g Brown - feeds Q terminal at the bulkhead. In engine compartment it goes to ballast resistor on a typical points system and connects with 14 g Blue feeding + side of coil. ( This circuit is active while cranking bypassing the ballast resistor to direct feed 12 V to the coil)
Bottom vertical Ign -16 g Dark Blue with tracer goes to bulkhead terminal N. In the engine bay it goes to your Regulator ( mounted on fire wall) that terminal connecting on the ignition side of the regulator also feeds the other side of your ballast resistor on a points ignition to drop the coil voltage while running.
By them soldering the ign1 and Ign 2 together they are effectively bypassing the ballast resistor. There are easier ways to do this. Shouldn't be a reason for hot light.
Vertical batt terminal - 12 g red - This is direct battery power. It is fed from the Battery to a junction on the starter relay then through the fuseable link to terminal J on the junction box. Continues through the bulk head up through the ammeter on the back side of dash panel then junctions off to the Battery side of the key switch, fuse block (battery side), headlight switch, and back out through terminal P on bulkhead to feed the battery side of Alternator.
I am not seeing any correlation to the temperature switch (unless I am miss understanding what you mean by hot light)
If you put the old switch back in does the light go away? I am thinking the new switch has a short between the battery and accessory or there are a couple of frayed wires that are contacting each other by the ignition switch.
Hope this gives you a start on your switch issue. Will not fix your running issue though. Get the switch working right first. Then on to the ignition.