Is it possible to pull some analytic info about the visitor that submits the form?

You could add some code to the HTML Tool possibly that has some sort of tracker. Nothing in the software directly does that, so it would have to be something you add to it.

Isn't there some kind of analytics built into the S-Drive when you have the paid plans too? Or is that not for the Form Builder stuff?

Overall there are plans to offer some more detailed stats for each service (Albums, Forms, Shops and Sites).