for point 2) I completed course work with high GPA, but had some difficulties meeting deadlines with the thesis work (now I see that my working ability dropped because of my condition). your examples sound reassuring, as I was afraid that he'll view my condition as a threat to the future productivity as a student/RA.
Well, unfortunately, course work is NOT thesis work. They are two completely different animals. It's good to do well in your classes, but a Ph. D. student needs to be able to do research and write. The thesis work would demonstrate that ability. Missing deadlines in the thesis work is not good thing. Needing to push back the defense because, for example, you spent a month in the hospital, is probably not a problem. Getting an extension because you can only work 4 hours a day, 3 days a week may not be a problem. However, having a history of seldom meeting a deadline will be a problem. I hope you fall into one of the first two categories.
for point 3) - I did not tell my adviser about my health problems before, and I was just wondering, whether I should explain my situation in detail, or just present him with the doctor's notice and ask how the extension is affecting his plans. I guess the latter is better.
Not telling your advisor about your health problems causing the need for the first extension was a mistake, but nothing can be done about it now. You haven't said what your health problems are (and we don't need to know), but keep in mind that some conditions are viewed in a more positive light than others.
It sounds like you do not have a close relationship with your advisor. You don't need to show him the doctor's notice (unless he gets really adversarial and requires it) and you don't need to give him a lot of details about your condition, but you do need to be upfront with a realistic timeline and amount of work that you can do.
For example, he doesn't want to hear that you vomit five times a day, but he does need to know that you only have the strength to work for about 4 hours a day and this will continue for at least six months, according to the doctor's best estimates.
I hope these clarifications help.