a. The crystal should only be connected to the dsPIC pins and loading capacitors, it you are also driving other circuits from the crystal connections the answer would depend on what those circuits are. It isn't recommended that you connect anything else to the dsPIC oscillator pins.
b. The oscillator does not run during programming. You can program the dsPIC with nothing at all connected to it except the Pickit.
Be careful how you check the oscillator is running, it is a fairly low power circuit and connecting an oscilloscope or frequency counter to the pins may be enough to stop it running. A better check is to load a simple program to toggle a pin and see if it runs.
Brian.