Yes, the do.
Their amount depends on your approach to learning. C1 still implies very limited knowlege of a language.
You can be MUCH better than C1. And if you ARE much better than C1 in some respects, then you may be much worse than C1 in some other respects and still have C1.
If you think that at a certain level grammar mistakes "dissolve" on their own, no. There are many grammar points you can ignore and this won't hinder communication. And this is what happens sometimes: some people ignore such points completely.