Hi -
So you know, this can be kind of hard.
First, to submit to a calendar, you'll need to write code or you can use qRules (it submits to a SharePoint list, which is what you need to do here).
Second, to check whether the date is valid, you'd want to query the list prior to submit and compare the start datetime with the existing entries to see if you have any overlap or duplicates - and display an error for the user.
So - I'm not sure what you have tried thus far, but maybe this will at least give you a clearer map of what needs to be done. :)