Sorry, can't help with the drawer front or the MW hole , except maybe put another MW in there.
If the cabinets are that filthy, maybe it's easier to strip everything back to the wood, fill any dings/holes, sand, use a wood sealer, prime and paint with a good cabinet paint.

Are they birch plywood or birch wood cabinets?
Since it's a rental, I would explore the option of restoring the wood, i.e. first cleaning and then using restore-a-finish.
What does the empty space look like above the wall oven? Could you just leave it or install shelves or wine rack?
For the drawer front, is it a standard size or can you look into ordering online, maybe Barker Door?

Here is a link that might be useful: diy restoring birch plywood cabs

The cabinets are birch wood cabinets. I had thought about making a shelf above the wall oven and a microwave oven count be slid into that space. Thanks!

I'd clean and try to restore the wood cabinets. In my opinion, renters are not very gentle, and wood will be easier to withstand than paint, which may chip.
I'd check on the woodworking forum for advice on how to proceed.