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We are not able to help you with this endeavor, unfortunately. This information is something that is best learned through obtaining a business degree. This would ensure that you have all of the proper knowledge before going into the task.

I am sorry that I am unable to help, but this is not something we deal with regularly.

Sometimes they can survive with 2, 3, or even 4, but that would be it. Do not pull them out, though. Just pinch the top to separate it. This will avoid disturbing the roots of the others that you want to keep. This article will help you with the care of these after the seedling stage: https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-carnations.htm

Yes, you can transplant them, but be very careful not to disturb anything. This article will give you more information on the care of these: https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-carnations.htm

If you deadhead them, they should continue to bloom. Remove spent flowers down to the next node (cut just above). If they are floppy, you can stake them. In the fall, cut them down to about 2 inches.

They need full sun, but they prefer a cool, mild climate. In looking at your photo, the green foliage without flowers looks more like chrysanthemums, which don't bloom till late summer or fall. Could you have those? The foliage doesn't look like carnations at all. However, there is a small pot in the center that looks like a dianthus flower that is blooming.

Here is more about carnations. Make sure you are meeting its growing conditions and they are getting full sun:

https://www.gardeningknowhow.com/ornamental/flowers/dianthus/carnations-in-containers.htm

You will want to germinate these seeds the same as if you were planning to plant them outside. Sprinkle the seeds over the top of starter soil and cover, just lightly, with more soil. Use a mister to keep them moist, and create a greenhouse environment for them by covering them with a plastic bag. When they've produce two or three leaves, they can be transplanted to a larger pot. This article has details:
https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-carnations.htm