What of a diver who simply could not control the nose and had to hold it?
I am not an instructor so I am not sure where the term "mastery" fits in, but I should think someone like this would be at risk of serious injury or death if they cannot operate uw without a mask. Sometimes you just need both hands to do other things.
Do I think they should have their certification withheld? I don't...but I think these people should really consider what they are doing and what could happen. If they don't practice these types of skills...it could kill them one day.
What then of a one-armed diver?
While we are not discussing the one armed diver per se, disabled divers who are not capable of performing the basic tasks on their own are required to dive with a trained buddy who can help them specifically with these types of issues, so really this is a moot point.
OP-
You do not have to like or be comfortable uw without a mask, but you do need to be able to function. As a contact wearer I will not open my eyes uw with my mask off either, but should I lose my mask then I will certainly sacrifice those lenses to be able to surface safely. If I were you I would find a low chlorine pool to swim in and open those eyes until it does not bother you anymore. Probably you will never ever have to...but the point of training is for that one time something happens.
As for your nose...I don't know what the standards say, but I can tell you that not being able to breathe uw without holding your nose can pose issues in a real life situation. If you are not able to close the passage between your mouth and nose try this:
Close your mouth and exhale through your nose, while exhaling rub the back of the top of your mouth with your tongue. Does that shut your passage off?
In order to do that move I have to contract the muscle that closes my palate. Not everybody can do this, maybe you cannot.
In that case I would suggest you think long and hard about diving. If you want to continue and your instructor will pass you, congrats and welcome to the club. However you need to be aware that you have a potential issue. As long as you know about it you should be able to minimize the impact.
Things like not diving in high current areas, or around seals, and being careful of where your buddies are so they can't knock your mask off by accident.
And make sure that you can operate all the gear you need to with one hand.
I don't think it is a deal breaker, but something more akin to a handicap. If I had 1 or both of those issues there are many dives in my log I would not have done...something to consider.
Mostly though you are complaining of something many new divers deal with, and time and practice will make these issues a thing of the past.
Good luck