Hello Ross

You should take a look on the LuxCal calendar : http://www.luxsoft.eu/    and here : http://calendarforum.dk/

This calendar has the possibility to make bookings. You can set a category to allow overlap or not allow overlap.

You can set a fixed duration for an event. You can set if a user should be logged in before he is allowed creating an event.

If the booking is for participating an existing event, then there is the ADMIDIO https://www.admidio.org/ .  Here you can have events, that users can participate. You can set a limit for participants and you can set a last-date for participation.

Both the LuxCal calendar system and the ADMIDIO member-system are open source systems and are free.

They do not have synchronisation with Google calendar, but LuxCal can automatically send reminders, and can also manually or automatic export calendar-files that can be imported in other calendar programs.

You could also make a script that took data from the emaoil-form and "transformed them into an event", but why not use a system that can already do that - and/or use a system that is already a web-calendar.

If it is for booking events I suggest the LuxCal calendar.

If it is for participating events, I suggest the ADMIDIO.

Or you could pay a lot of money to buy a system and afterwards adapt the system to your need.