I think Dan is correct with the seat switch - - but just a guess on my part too.
The tractor "makes sure" there's someone in the seat via a simple compression switch in the seat. When a person sits in it, switch is compressed and all is good.
Applies to forward/reverse movement, as well as operation of the PTO in most instances.
Guessing when you're backing up a hill, you're taking weight off the switch, and it causes the engine to shut down. Sounds like the switch is doing what it's supposed to do. Not sure the issue with the PTO, but could be related to the same switch.
My first suggestion is getting an operator's manual for your tractor. While you have a BX1850, probably any BXxx50 series manual will be helpful. If they have a Workshop Manual (WSM), that would be helpful also.
Here's a couple places to start.
Kubotabooks.com
Kubota Tractor User Manuals Download | ManualsLib
Also - - parts diagrams with part #'s can be found here:
Kubota Tractor Corporation - Illustrated Parts List (kubotausa.com)
Switches of the type can not be "reset," but can be tested for their operation with a multimeter.
There are other ways to go about testing.
That said, in your own words you just endeavored on your first and only maintenance project when you replaced fuel filters. That's routine maintenance.
I'm reluctant to instruct you how to bypass safety features for testing purposes. No offense intended, just don't want you to get hurt.