Car alternator broken soldering joint
So my question, is it normal to have no resistance reading on voltage regulator while it isn't connected to 12 V or do I have two problems (my voltage regulator died and I need to re-solder that joint)?


high impedance is expected for linear regulators without bias.
Solder joints worked in -40 'C in Winnipeg so they should work where you live.  Just fix that with resin-core solder and check the rest. (marginal defect)

It appears you only lost 1 of 3 phases but that reduces efficiency to less than 1/3 max power for delta windings. I guess the car may have detected/indicated the fault or perhaps you did with the rear windshield heater and headlights or dead starter.
p.s. in my retirement, I moved to warmer climate.