Is non-degree studies a viable path to a master's degree? I had a very rocky undergraduate math career that ended with me having around a B- average due to undiagnosed/untreated anxiety issues. This means my final GPA is around a 2.8.
Would non-degree studies be of benefit for admission to a master's degree program if I do comparatively better in my studies and develop a working relationship with professors whose work interests me? I am leaning towards an economics or a mathematics master's program at the moment.
Also, will my diagnosis hold me back during the admissions process should I decide to pursue an advanced degree after satisfactorily completing non-degree studies at the graduate level?
The things hindering you would be both your level of preparation in required subjects and the level of competition in any place that you apply to.
If you are missing certain courses then you can probably use non-degree status to make up and get a formal transcript. For repeating courses you have, it might work out as well, provided that you do much better. But if you have adequate knowledge even with low grades it might be better to take more advanced courses rather than repeat.
A medical diagnosis shouldn't impede you in any way in the US. There are laws protecting people with conditions, including privacy laws.
But, you can proceed in parallel, actually, signing up for a course or two as a non-degree student while also applying to a few masters programs to see how you fare. The more competitive the program, the lower the chance of admission, of course.
Another option is to just talk to an admissions counsellor at a place you'd like to attend and see what they recommend. They might have you correspond with a math prof. But, in general, the more information you get the better off you will be. Don't proceed down a particular path based on assumptions.