I usually use a common IRLZ44N but yours should work ok.
A gate resistor of 220 ohm to the Arduino and a 10k to ground.
Separate supply to to motor will give some noise isolation.
When choosing switching device, remember motors may require up to 6 times full load current instantaneously.
What voltage split are you aiming to get and at what load current?
Do you have a link to the specs of the board you show?
Point being it may be possible to use just the one board, run to output up to 30v (for say a 15v split) then add a couple of linear regulators positive and negative along...
Well, the word synchronous and the frequency 50/60hz say it is which is 2:1 against the 12v DC.
I've seen worse with stuff from PRC.
Every image I could find in relation to 12v synchronous motor TYC-50 say it is AC.
I'd tend to go and find a "real" synchronous motor @110v mains rating and use...
Some dishwashers will run the motor after a certain amount of filling water is detected (via pressure switch or time delay) to begin internal circulation.
To drain they run the motor in reverse to pump out, so perhaps fault lies somewhere along those lines.
Without any schematic or model/make...
This IS star-delta starting or commonly known as reduced voltage starting.
Usual configuration is bring out the six coils cables, connect in star for starting, then switch to delta for running.
There are uses requiring star connected run but not so common.
Current is limited during the run up...
Battery recharge current is a factor of battery size and type.
Too many factors for any open ended answer.
Regardless to say it has next to nothing to do with the power outlet in your house or wherever.
Seems ridiculous then for manufacturers to cut-in and cut -out via a simple thermostat.....think you need to re-visit auto air con 101.
Any required protection is already built in to a working system.