Did the deities live on Mt Olympus?

Answer 1

That is what the myths tell us.

Only the Olympic Gods "lived" on Mt Olympos in Greek Mythology. There was a constant need for the balance of Olympus to be maintained, so additional deities were usually ordered as such, including Hestia, Goddess of the Hearth and Home, who was eligible and rightful to Olympus, but having upset the balance of Gods to Goddesses, she was not considered an Olympic God, and gave it up on her will, as to her it seemed more vital to dwell in the hearth and home.

And even the Deities didnt stay on Olympos all the time- Artemis, the Goddess of the Hunt, was always one to go on hunts in the wilderness,and represented hunting and the movements of animals in these aspects, Apollo had a holy place near Delphi,Poseidon dealt with the seas, so often he was in the sea, and representative of the sea.

Often all the Deities were not assembled unless it was a full-fledged call of Zeus to "come to order".You can also see how often the deities were not constantly on Olympos, rather being representative and of what they were.

Answer 2

Yes