MK160 Rating

Can the MK160 be used to directly switch on a 120VAC 125 watt engine heater, ie without having to go through a relay/solenoid switch of some sort? Thx.

1 amp should not be a problem.
Make sure to put the unit into an insulated enclosure and make sure all AC wiring is performed according to ‘good engineering practice’.

Is an AC adapter (12V DC output) rated at 50 mA adequate to power the MK160?

No, you need at least 100mA.