How do you recommend cleaning your stainless range hoods? The outside especially is very greasy, with a nice mixing of dust caked into it. Last owner obviously didn't keep it up!

It needs servicing. They should be done once a year, and especially just before the summer season. Yours has a clogged condenser grill full or greasy dust and fluff. That is causing it to overheat and will cause the compressor to fail.

Cleaning cycle should be done on recommended intervals, plus additional bleaching inside the ice storage bin, especially up in the top where you can't see.

Clean Greasy Range Hood Filters How to Clean Your Stove Hood Easy Cleaning...

read the mixture instructions
try preheating the oven for the required temperature or when the thermostat turns the light off
insert the cake to be cooked and leave it there for the desired time
Too hot will burn the outside with out cooking all the way through
check the ingredients carefully as if the mixture is too wet ( butter, milk, water all make it wetter than required )
Place the cake in the center of the oven ( not on the bottom shelf or the top shelf)
All these may sound mundane but they all point to getting a good cake result

!2VDc or 12VAc? If Dc check for a loose, corroded or greasy ground connection. Also check the center pin on the light sockets, they get flattened and coated, a good cleaning may solve your issues.

If your microwave is ducted to the exterior the cold air could be from the outside if the backdaft damper is not functioning properly. If you have a HVAC system, the return air process can be pulling outside air through the hood making the condition worse. The microwave damper is usually on top of the microwave unit at the exhuast opening. Maybe some greasy dust is preventing the damper from closing all the way.

cream of tartar moistened with water and a paper towel

You will need to replace the temperature probe in the oven. Here is the part to order.

I took the motor out of the range hood. There is a set screw holding the squirrel cage fan in place which I unscrewed enough to slide the fan part off. I then boiled the part in a deep pan with lots of dish soap. I then brushed away the remaining debris. Avoid using dishwasher detergent which is effective but seems to erode the aluminum.

vinigar might work

try using stainless steel magic we use it here it works good