How long had the cap been on the bench "charged"? That sounds pretty unusual, it's more likely the PIC was damaged due to a wiring short etc.
The PIC input pin is already protected to some extent by its internal pin diodes, these are good for 20mA continuous and a couple hundred mA for short term one-off events. I don;t think a 10uF cap charged at 5v has enough energy to blow the PIC pin diodes. Maybe if your 10uF cap had been charged to a higher voltage it could do it.
You can add a zener if you like, and the increased initial capacitance is just because the zener adds a bit of capacitance, as you measured about 65pF. That should zero out normally when you press the zero button.
The 100 ohm resistor in series with the RA0 pin is also fine and should not really affect the operation of the meter, and you could even to go 330 ohms there for more safety.
Personally I'm in the habit of shorting cap leads to discharge them before measuring IF they have been in a breadboard etc and may have some charge left in the cap.
And if your 10uF Mylar cap was a high voltage type, that you know had been connected to a high voltage, that's a perfect example of WHY why you short out cap leads before measuring, even if the capmeter does have added protection!