Hi, welcome to BYC!
Congrats on your hatch.
I probably won't be much help... but I will tell you what I do anyways..
Well for incubation.. I use turners and a foil water boat so I just have to switch out the boat and never clean my wells, since I run "rolling" hatches. The wells do tend to get slimy... hence the switch and not using the bottom ones are harder to clean.
In my "hatcher" I line the bottom with paper towels and shelf mat, still using foil boats or wet sponges for my humidity. After hatch is complete I roll it up and throw everything away. It usually catches it all. At most, I rinse and wipe the bottom wells if any water got in them. And I use a wet paper towel to grab any left behind egg shell or wipe the incubator walls. Then I set up again and get ready for the next hatch that usually comes within a few days.
As far as fridges go... In the kitchen... we use bleach or whatever and never worry about the electrics as it seems mostly sealed. I wouldn't wash the light bulb or use anything wet on the bulb when it's hot. As far as residual staying behind and effecting embryo's I have ZERO experience there.
I's funny... I'm a MAJOR over thinker... but some things I am able to keep it simple. In no way am I suggesting that not cleaning would be appropriate. But what I am saying is that I go chemical free as much as possible and am still able to maintain good viable hatches. So I guess I wanna say don't worry too much.

But I know it's important also!
Hope you get better answers from someone who has better procedure than me.
Oooh, a fridge size incubator!
I would be in some trouble.