We still need to know what is going to be connected to the PIC. The PIC itself shouldn't need much current at all, but most likely that's not all you are doing with it.
More than enough! A 4x4 display would use around 160ma. Make sure that you have a 5v output, a PIC can run on 2V to 5.5V, so make sure it is well regulated for best performance.
I bet the power supply is a cheap unregulated one. Its output voltage is probably about 9V or higher at a low current. It will fry the PIC without a voltage regulator.
Does it charge through USB? If so, not without the regulator. The USB Vbus is regulated, so if you are trying to emulate power from a USB port you need it to be regulated.
If it charges through a different jack, it is impossible to tell without knowing details about the camera.