I felt the same way in graduate school and so did a lot of my peers. At the time if felt like the professors thought it was their 'job' to demean us. Looking back, I see that it was their job to help us learn to think at a higher level but some of them were nicer and better than others than getting this task accomplished!

I don't think it would be fair or appropriate to ask your professors what they think of you or your performance - that will be reflected in your grades. Asking for their guidance with course work is appropriate and you can certainly do that without causing harm,

Anyway, as you said in your last sentence, 'no amount of work will ever make you qualified' is close to the truth ... it is the work you are currently doing that ultimately will make you qualified but you aren't there yet.

Get some support ... peers, friends, therapist ... grad school IS hard.