I suspect that if that charging socket on the phone is faulty then the phone will not charge on the mat either. The charging mat is still dependent on the phone's existing charging socket. You simply plug in a little charging unit (about 5cm square on a 10cm cable) into your phone and then place that unit on the mat.
I'm afraid I can't help on that point. I sed it with an iPhone and found it fast and efficient although a little finnicky at the beginning whilst I worked to find "the spot". After a short while I knew where the spot was!
If you have the case for your phone that is compatible with this wireless charger then there is no reason why this shouldn't work, however if you just have the standard note 2,it does not incorporate wireless charging, and the official Samsung rear of the phone with wireless charging is qi standard. This wireless charger is not qi standard and therefore not compatible with qi receivers.
I don't really know the answer to your question. Is the Galaxy Note 2 already enabled for wireless charging? If so, then I expect that it will be "QI" compatible. This appears to be the universally accepted standard. However the power matt is not QI compatible and will not work with QI compatible devices. I have a Nexus Five which would not work with the power matt device. However, Power mat do make some receiver devices that can be fitted to some removable phone batteries, so you may be able to adapt your phone to work with the Powermat but you should enquire directly with Powermat to see what is available. If you phone is already QI enabled, then you should buy a QI compatible charger. There are many different manufactures and models available. Do a search for QI Compatible chargers. I hope this is useful.