JScheduler is just the tool for that, it's a pretty ease to user Joomla! component: the visitor gets to see a calendar and a contact form. He / she choses an user to schedule a meeting, selects a date and a time, fills in his / her data and sends the message with a few words about them and why they want to meet you.
In this way you and your team can organize your schedule better.
In the backend you can choose what days of the week is each user available, the time and the length of the meetings / classes / whatever, you can asign Joomla users to this component - the involved users will be have to configure their availability in order to get messages and the synchronization with Google Calendar. You can use the calendar in the backend instead the front-end form to set meetings with the members of the team or reserve some slots for private or public events (the public ones will be listed in the frontend for other people to know and come).
You can't delete the messages you get using the frontend, but you can set multiple meeting sessions (with you, the visitor and / or members of your team - maybe their presence is required or the original message was sent to the wrong person), asign notes to each meeting and delete future schedules if they are not going to happen.
An example is here, the company's website, and you can user JScheduler to contact us.