Given the recent photos, you have a 2010 model. I believe that is when Montana introduced the dropped frame and the convenience center. The description I provided earlier about the tanks and dump handles is not correct for your year.
The black and shower tanks run longitudinally, parallel to the frame. The kitchen tank is as I described previously. Perpendicular to the frame above the axles. The black and shower tanks have flexible cables that operate them. The operable end is there in the convenience center. The cables run to the back end of the tanks, where the valves are. It is common for those cables to get corroded and hard to move. And likely the valves and its gaskets are shot. They are not that difficult to replace if you are handy. A Search will uncover discussions about the process.
The kitchen tank has a 1 1/2" drain line that connects to the 3" drain pipe from the black/shower tanks. You can see it in the latest set of pictures you provided. The valve for the kitchen tank is, by code, installed close to the tank. It has a solid rod that passes through the frame and directly operates the valve.
1) how urgent is it for me to replace the one stuck open gray valve,
This one can stay open as long as you have a quasi "P" trap (i.e. a dip) in the sewer hose to prevent sewer gasses from coming back into the tank and RV.
2) the hard-to-maneuver black valve,
This needs to be fixed regardless of it is open or closed. If closed, then it will fill up and then you have problems. If Open, then the liquids drain and the solids build up and solidify. AKA the the poop pyramid. Yet another problem you don't want.
3) How do I officially determine if my black tank is all flushing or not/
The best answer is a clear extension attached to the sewer pipe. It gives you a clear picture of what, if anything, is coming out the pipe. When flushing the black tank it helps you to know when the output is clear and clean (as much as it can be).
Amazon.com
whether I need to repair something/or if I can accept sensors that don’t work great and just flush the black tank when it gets to 2/3, even if it only goes back to 1/3, and finally
The sensors are notoriously inaccurate as you have learned. There are alternative sensors that live outside the tank that you can install - Seelevel Sensor
https://www.etrailer.com/Smart-RV/S...vJRiXr0aRKs01_l6OKyfw8KEBtqohMoBoCcCsQAvD_BwE
Most of us just get used to how frequently the tanks need to be drained based on our usage. The toilet tank burps when close to full and the shower backs up when its tank if full. You don't want them to do that very often as it is hard on the tanks.
4) Where the heck is the second grey tank valve??
The handle to operate the valve comes through a hole in the frame or coroplast (depends on the model) somewhere around the wheels. It is not uncommon for it to be missing. Drop the edge of the coroplast and you can see the valve. The rod is 1/4" thread and easily replaced. I am pretty sure CW has them along with other places.
Is this the second gray tank valve?
That is a bracket that supports the kitchen drain line. The valve is further towards the rear where the tank is located.