Is it possible to have a partial dry socket?

Not really. It is either healing normally or not. You either have a dry socket or you don't, although there are degrees of pain, different pain thresholds, etc.
Of course. A dry socket can either be partial or complete. It depends if only a portion of the blood clot is destroyed or the entire blood clot is affected. Regardless, if the pain persists after 48 hours, return to the dentist for further assessment and dry socket treatment.