1. Direct hotwiring of the fan to the battery *WILL* drain the battery when the car is not on, verify that the fan is NOT hotwired to the battery directly and is instead wired through the ignition by checking under the hood after you have turned the car off. You can hear and see the fan turning in most cases. It should be evident when the car is running and evident that it is not running when the car is not.
2. Crud in the radiator will, in general impede your cooling system and it can clog or otherwise damage a thermostat. This, however, will generally cause the engine to run hotter, which should cause the fan to run more, not fail to run. I think your problem is more likely the coolant temperature sensor (there may be two, one for the gauge and one for the fan, so you'll need to double check this) rather than the thermostat. If you replace this, and un-hotwire the fan, you should at least have fixed the electrical issue. I would recommend, however, REPLACING any leaking component (radiator, etc.) and using a cooling system flush product to clean the Gunk crud out of the system, because:
3. Yes. That crap is horrible. It may stop a radiator leak, but it also will block passages in the radiator and especially in the heater core, which on many vehicles is a real pain to replace, so I would never use radiator stop leak of any kind. It can screw up the thermostat too, and if that's the case, no manner of fan hotwiring will keep your engine from overheating.
Next steps: replace the radiator, if it was leaking and "fixed" with stop leak, it's a bad radiator, sorry. replace the thermostat, probably ditto. I'd replace any hoses associated with the cooling and heater system as people don't replace these nearly often enough, usually wait until they leak, or blow completely, and then you have real problems, so I'd do this now while it's easy. Then I'd replace the faulty temperature sensor - your parts store should be able to direct you to the right part here. THEN, I'd have your mechanic friend un-hotwire the fan. Everything *should* be okay now, except that there is still gunk floating around in places that are difficult to impossible to get to, such as in the cylinder head, engine block and heater core. Not much you can do about that, and I wouldn't recommend replacing anything unless it's completely unworkable. Again, flush and fill, try to clean it up as best as you can.
Good luck!