To create a field that requires a user to select 10 choices, you create a multiple choice field with your 15 choices. Then in the Field Properties pallet check the checkbox labeled Limit Choice Selections. This will expose a couple of input fields for the Min and Max number of choices a user has to select before they can submit. Put 10 into the the Min and Max fields to require exactly 10 choices.
As for limiting a user to only submit once, this is functionality that is not supported by FormsCentral.