Yes, you would need to re-submit your transcripts. As for your letters of recommendation, you have a few options. You could either (1) keep in touch with the letter writers and have them re-submit the letters next year, or (2) ask your writers now to send the letters to Interfolio, a $49 (I think?) service that holds your letters for you, so you can send them next year without having to pester your recommenders in the future. It's a pretty nice option, really.
The MCAT is generally offered Jan-Sept, so October isn't an option. You can tailor your study schedule however you want, but you should plan to devote 3 months of full-time effort (or the equivalent of that) to studying. If you're working and volunteering concurrently with studying, you may need longer than 3 months.
The safest option would be to plan to take it between January 2022-April 2022. That way, you will have plenty of time to study, and your scores will come back before June 1st, 2022, so you will have peace of mind knowing your scores, and can submit early.