I want to know if I should email my new boss and say, "thanks for being so patient with me". I'm an excellent worker, but I made a few mistakes (and I haven't even started my first project yet). I have some mental disorders, which was the cause of the small, minute mistakes. In any case, should I email him and just say, "hey I just wanted to thank you for being so patient. I'm a pleasure to work with, and I appreciate you for (what should I write here if so)" ... or should I just do nothing?
You've said yourself that the mistakes were small. If there was a real problem you boss would have talked to you already.
Do nothing. Make a note of the circumstances that caused the mistake so that you can spot when they are going happen again. In this way you can hopefully prevent the circumstances occurring in the first place or take extra care when they do to avoid making mistakes.
Everyone makes mistakes from time to time. Don't dwell on your past mistakes. Learn from them and move on.
Yes, I know that this is easy to say and hard to do, but it doesn't hurt to be reminded of it from time to time.
While I agree with ChrisF's answer in that there's no need to point out your mistakes (minute or otherwise), and your focus should be on avoiding the repetition of mistakes, I would like to add that as a boss I really appreciate it and take note when a newly-hired employee sends unsolicited feedback about their own performance (especially when it's new job), and looks to be proactively trying to improve on their own. However, that being said, there's a fine line to walk.
Some not so good examples (in general):
Some perfectly fine examples (in general):
A message that's always welcome from a new employee: "Just wanted to say I appreciate the opportunity and enjoy working with everyone. I look forward to contributing more!"
Looking at your specific message:
Two comments: first, pointing out that they're patient is just fine, but you probably want to avoid doing it in a way that implies you are high-maintenance. Second, I wouldn't say that you're a pleasure to work with -- you may very well be, it's true! -- as that's something your boss will find out/come to his or her own conclusion about, and also, they hired you so they probably already assume you are (no need to add doubt into the mix).
So, something like "Thanks for your patience while I get up to speed in this new position" is totally fine and doesn't say too much one way or the other but lets your boss know that you're paying attention both to your work and your work in relation to the company/your colleagues.