Yes! Please specify on the RSVP that you will be including a plus one. The invitations will say "and Guest" so there shouldn't be any confusion.
If your kiddos' names are listed on the RSVP card, then yes, they are invited! If they are not listed, plan on enjoying a kid-free evening!
Not for us. We will be enjoying our wedding evening with our little family at home. By all means, feel free to keep the party going!