I haven't really experimented enough to verify to myself that the following is true so I am relying on what Ansel Adams and Polaroid have written in years past to be true. In general, overdeveloping does not occur or, if it does, it is a slight amount. This is because the chemical reactions taking place are structured so that they play out after a little over a minute or so.
In my own verification, I always develop Type 55 negatives for a full minute, contrary to the instructions. Nothing really changes going even a bit longer.
I have intentionally tried this with Fuji and your mileage may vary. But I kind of doubt any real harm will come from overdeveloping.
The color does not develop to completion. It alters the color balance to under or over develop. The B+W stuff does AFAIK because you can leave it for up to 15 minutes according to the fuji instructions.
All peel apart instant products stop developing when you peel them apart. They have an acid layer and a timing layer to reduce alkalinity.
All integral instant products have a timing layer that kicks in roughly 2 - 5 minutes after the pod bursts and this brings the pH down which shuts off development.
All instant products, just like all normal photo products are temperature sensitive and the results vary with temperature.
So, the answer is "NO" they do not develop to completion, but they can suffer from temperature effects which can over or under-develop them.
If you are outside in sub-70f temperatures, just put the developing picture under your jacket in your armpit. This is an ideal environment for adequate development. Just to clarify, outside your shirt but inside your sweater or jacket in your armpit.
It depends totally on what temperature the film itself is at. If you have had you camera and film exposed to cold temperature for a long enough time, its going to take a while to warm up. If it freezes, you make likely burst the chemical pod. Instant film should never be frozen.
Its best not to try that. Instead, try keeping the film warm until you fully ready to expose the shot. Then do something to keep it warm while it develops.