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Yes, it can harm your seeds and eventually kill them. Start by decreasing the humidity around the seeds by removing the cover or poking holes if there is a bag around them. That should prevent the fungus from growing futher.
Here is more:
https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/preventing-white-fluffy-fungus-on-seed-starting-soil.htm

It can eventually kill the seedling if it isn't controlled. High humidity is usually the culprit. This article gives tips on how to stop the white fungus.

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/preventing-white-fluffy-fungus-on-seed-starting-soil.htm

Just put it under the soil about half an inch, to an inch. Plants can detect gravity much better than you would think. This collection of articles will help you to grow peppers properly: https://www.gardeningknowhow.com/edible/vegetables/pepper

All peppers can benefit from topping. They can be treated as perennials (even small trees) indoors. This is a way to keep them alive for many years, as long as they don't get cold.

Yes, even more so than those grown under light. They will not be used to the high intensity sunlight, since your windows will filter out most UV or drying winds. Start them out with about 4 hours of outside time per day for about a week. This will be plenty of time to harden off. Starting high light requirement plants in a windowsill will usually cause extensive stretching to the point that they may not hold themselves up upon maturity. I recommend starting under LED lights before I would recommend starting in a window. They stay much more compact under artificial lighting.

This article should help:
https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/seedlings-turning-yellow.htm