Hey there, EDHAMDUG,
Thanks so much for using Apple Support Communities.
We want to help you get Calendar working at its best for you. To start, we want to make a note that the gray dot you see in the month view in the Calendar app simply indicates that you have an event on that particular day. This dot doesn't indicate that you are busy during that timeframe but rather just that you have some sort of appointment or event that was created for that day. It could also appear for an event that was created by a calendar you're subscribed to such as the default holiday calendar for your region.
Additionally, if you'd like to create an event but not have yourself marked as busy during that timeframe, you can follow these steps:
"You can add an event to your calendar without having the timeframe appear as busy to others who send you invitations.
- Tap the event, then tap Edit.
- Tap Show As, then tap Free."
That info can also be found here: Send invitations in Calendar on iPhone - Apple Support
We hope that this information is helpful.
Have a great day!
