Yes, you should be able to check the wires going into the R/R.
I would have your battery load tested if it was dead, it might show 12 volts but no amps. A DIY test would be to charge the battery, connect a meter to it and it should show around 12V, crank the bike and watch the meter, with the starter turning the bike over, the voltage shouldn't drop below around 9.5V, if it does, the battery is bad. 13.5-14.4 doesn't seem bad.
Yes it can, you can connect a meter to the battery and monitor the voltage while you use a heat gun on the R/R just don't overheat it. A can of spray duster turned upside down can be used to cool components if checking for a cold failure.