Going to the comparis website, which is mentioned in the first sentence of this FAQ, is the best way to find out which insurance companies might be good for you (i.e., it will give you a list with company names and prices). On that site, click "En" for English in the top left, then click "Health Insurance" in the panel on the left, then fill in the details requested.

Regarding dentists: Yes, they are quite happy for you to go there and pay them directly for whatever work they do.