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As long as the ground has not frozen where you live, you can plant them and they will bloom. If it has, this article will explain how to store them till spring: https://www.gardeningknowhow.com/ornamental/bulbs/bgen/preparing-bulbs-for-winter-how-to-store-bulbs-for-winter.htm

No, these bulbs are pretty much self cleaning. They need their leaves to store energy for next year, so hold off a bit before cutting them if they are naturalized in your lawn. As for the flowers, you can take them off if you like, but you do not need to. They may go to seed and produce more crocus. In some bulbs, the seeding can take away from the flowers for next year, but crocus are so prolific that it is not a concern for them.

This article will help you with that:
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-plant-a-flower-bulb-in-your-garden-after-winter-forcing.htm

You can take the faded flower, but leave the leaves. They need those to store energy.

Either there is a phosphorus deficiency in the soil or the corms are overcrowded. Try giving them a high phosphorus amendment or fertilizer (like bone meal) and dig a few up to see if they are too crowded.

Only in very warm areas do they need to be dug up. Most places do not need to dig them up in the winter to pre-chill them and they actually need cold weather to bloom.