Watch and wait. It is very difficult to estimate the amount of tissue that will necrose (die) as a result of frostbite. Conservative treatment with gentle rewarming and observation is typically recommended. Removal of dead tissue is performed only when it has fully demarcated.
Hopefully not. Get seen asap. Gentle rewarming is the best treatment after you can be sure no further cold damage will occur. Any surgery is done only when the depth of the injury can be fully judged (i.e. Not right after rewarming).