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Yes, I would recommend wating for the blooming to stop before repotting the plant. Also, if this is one of the hardier types similar to what you have growing outdoors, then yes, it would be ok to plant it with them.

Yes, that is very unusual. Since it's doing so well, I wouldn't change any of your cultural practices, but when the flowers start to fail, you might want to let it go dormant for a couple of months, then see if it will regrow leaves. https://www.gardeningknowhow.com/houseplants/cyclamen/cyclamen-care.htm

Did your cyclamen recently bloom? After a cyclamen blooms, it will go into a dormant state and the plant will look like it is dying. The leaves will turn yellow and drop off. So I hope that is what is happening here, that it's not dead, just dormant (sleeping). Once the leaves die, you should stop watering the plant and place it in a cool, somewhat dark place and let it sit for a few months.

For more information on the care of cyclamen, please visit the following link:
https://www.gardeningknowhow.com/houseplants/cyclamen/cyclamen-care.htm

Usually the spent flower will flop over. You can part the leaves and cut out or snip with a small scissors the spent flower.
This helps the newer flowers emerge.
This applies to indoor and hardy outdoor plants.

https://www.gardeningknowhow.com/houseplants/cyclamen/cyclamen-care.htm
https://www.gardeningknowhow.com/houseplants/cyclamen/hardy-cyclamen-outdoors.htm

You could but it would be difficult and time consuming to say the least. Most are only sold as plants. This article will help with growing them: https://www.gardeningknowhow.com/houseplants/cyclamen/cyclamen-care.htm

The tropical can only tolerate temps down to 40 F, while hardy can tolerate temps to -10 F. What variety you have and how cold it gets where you live will make a difference in whether you can grow them outside year round. For more information on its care, this article should help: https://www.gardeningknowhow.com/houseplants/cyclamen/cyclamen-care.htm