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They will likely not survive if they are exposed to frost. You can cover them with a sheet to keep the frost off them and this will help keep them alive. This info may also be helpful:
https://www.gardeningknowhow.com/plant-problems/environmental/frost-how-to-protect-your-plants.htm

You should seed the seed sprout in 7 - 14 days. The warmer they are (but not hot - less than 75F), the faster they sprout. Many people will place them on the top of appliances, like TVs or fridges as these stay just warm enough to speed up germination without overheating the seeds.

I would wait until you put it outside. Overwintered peppers are stressed at this time of the year and repotting puts more stress on them, which can harm them. I would wait until they are placed back outside. Then they will be in active growing mode and will be better able to recover from the stress of being repotted.

You don't need to dry the seeds to grow them. You only need to dry them if you plan on storing them. You can plant them as soon as they come out of the fruit or vegetable.

Now would be good. They will need them once they have their first true leaves. I would use a water soluble fertilizer at half, or even quarter, strength though.