When measuring ohms, a single "1" on the left side of the screen means an open circuit. It is the same as you see with the probes not touching anything. This is usually good when you are checking for shorts.
A one piece ignition coil mounted next to the flywheel is self-powered. It doesn't connect to the battery system at all. The only connection to the controls is the single "kill wire" which will stop the engine when the wire is connected to ground. That happens when the key is turned off or if one of the safety circuits activates, such as having the blade engaged with the seat unoccupied.
You can easily find out shorted regulator by leaving it disconnected and see if the fuse holds and the rest of the mower works.