Hal Higdon has a ton of training plans. Here's his one for multiple marathons.
The specific answers to your questions is to do what you want while listening to your body.
Higdon's plan for two marathons two weeks apart has a 1 -2 hour run on the weekend in between. That could be an half-marathon. It's hard to tell before your marathon if your body will feel up to a Half a week later, but if it is, a Half sounds like fun.
If you have time to keep up with the long runs while still doing shorter runs in the week, I would recommend keeping up with them. You don't want to be doing 2 -3 hour runs on the weekends and nothing during the week. And then when you sign up for your next marathon, you're not starting from scratch.
You'll want to sign up for your next marathon about an hour or two after you cross the finish line. Go for it. Read the multiple marathon training...