If you don't get the invoice, it means that they don't want to take any responsibility of your surgery, you can't sue them.
Go back and get your invoice, this is your right.
据我的经验,应该是统一收费标准的。
This is not 100% correct. If there is one small hole in your teech, he/she can say there is a big one and charge you much more.
The clinic is like the garage, you can't trust them. Last year, I went to the Chinese dentist in NDG, she wanted to charge me over $1000 in cash to fix my teech. Then, I went to another dentist, just $200 by Master Card, everything is fine.