Can I wash what's under the hood?

2015 Mazda3
I was told yes. It's very dusty. So should I spray an all purpose cleaner (I found one on the automotive isle at Walmart)? Can I then spray water all over what's under the hood? I was told yes since my car is Japanese and its computers are internal. I just worry about the wires and connectors. I was told it should be fine as long as I have my engine running. Really?
2014 Mazda3 S GT auto, 2008 MX5 6-speed
I would not use a hose but I do use a spray bottle to spritz as much as I can while waiting for the engine oil to drain. I also:

Check the battery to see if distilled water is needed.
Check all fluid levels in cooling tank, windshield washer tank, power steering tank, and brake reservoir, and replenish as required.
Apply Armor-All to hood and door gaskets, as well as rubber hoses.
Use non-glossy cleaning wipes on plastic surfaces in the car.