There's no polite way to say it. But if you must, just say adults only and call it a day. But be prepared for someone to bring their precious angel anyway...
That's fine. My parents have a deck, and it's not that cold in November. I'm sure it can play with the dog.
We requested no children at our wedding and we put "This evening is to be enjoyed by adults only" at the bottom of the invitation. You could change it to "This afternoon is to be enjoyed by adults only" Be prepared to stand your ground though, we had more than one phone call asking to make an exception for their little angel.