If I’m removed from a family plan can I join it back?

See the 'If you can't accept the invitation' section in:

Set up Family Sharing - Apple Support

"If you can't accept the invitation, see if someone else joined a family with your Apple ID or is sharing purchased content from your Apple ID. Remember, you can only join one family at a time, and you can only switch to a different family group once per year."

If that doesn't apply you can try to accept the invitation on the Family Organizer's device. See the 'Invite people to join your family' section in the link above.

"If your family member is with you, they can enter their Apple ID password on your device to accept the invitation."

Yes, the Family Organizer should be able to re-invite you.