Is it possible to sign up for a roster during an active Monster Outbreak?

Yes, it is possible. You can sign up for a roster by tapping on the outbreak point when it is no longer within your access range. It is necessary to sign up for a roster in order to receive the limited hunter medal. 

What happens when an outbreak ends while I’m still during a hunt?

If you don’t successfully hunt a monster within the time limit of a monster outbreak, it does not count towards the slaying goal number.

I have fulfilled the conditions for obtaining the limited hunter medal, but I did not receive it.

To receive the limited hunter medal, you must sign up for a roster and slay at least 1 target monster within the time limit/at the point you signed up. When conditions are met, you can receive the limited hunter medal by going to the main screen after the outbreak concludes.

Is there a way to participate in the Monster Outbreak alone, instead of a group hunt?

Monster Outbreaks will always become a group hunt. If you become the only hunter in a lobby, it is possible to begin a hunt alone.