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Updated Jan 13, 2021
Version 7.0
Blog Post
Just happened to find information regarding this so I thought to post it here also.
There is a lot of useful information about the functionality in https://docs.microsoft.com/en-us/microsoftteams/tmr-meeting-recording-change#permissions-or-role-based-access
It seems that channel meetings are the way to go to make sure that the recording lands in to the correct place. If the meeting is a channel meeting the recording will go to sharepoint regardless of who presses the button.
There is information regarding the timeline https://docs.microsoft.com/en-us/stream/streamnew/classic-migration#migration-phases
Your quota question is answered https://docs.microsoft.com/en-us/microsoftteams/tmr-meeting-recording-change#frequently-asked-questions