Some states will allow you to sit for a hygiene licensure exam as a dental student and if that is true and those states also allow hygienists to perform anesthesia, then you may be able to do some of these tasks.
You, under no circumstances, should be performing irreversible procedures as a dental student outside of the "walls" of the dental school (unless you are at an off-site clinic associated with the school, extenship, or medical mission).
In fact, I will run as fast as possible away from this dentist and his/her practice. When you are ready to graduate, don't look at this practice as an associateship. If the dentist is **seriously** asking a dental student to practice dentistry outside of dental school, I would not want to be anywhere near him/her. It smells of pressure to delegate assistants to work beyond their scope and further problems down the road.
If you were caught, at least here in FL, you would be convicted of a felony, probably kicked out of dental school, and banned from dental licensure for a number of years.
Don't do it.
Good luck.
-Mike