Is there a place I can check (other than my own bank account) to see when the State taxes I owe were (or will be) paid? I had requested to pay the taxes owed online.

check your state website to see if they offer this info. other than that it's only the bank account

How do I review the method I chose to pay my federal income tax with, after filing with TurboTax Online?

You can verify the tax payment method you chose on the Electronic Filing Instructions for your 2021  Federal Tax Return page, next to Balance Due/Refund. To see the page, follow these instructions:

1. Open your return if it’s not already open.
2. On the [Your name], your 2021  federal return was accepted! screen, scroll down and select Add a State (don’t worry, we’re not adding a state) to access your tax forms and worksheets.
3. On the left side select Tax Tools then Print Center.
4. On the TurboTax Print Center screen, select Print, save or preview this year’s return.
5. On the next TurboTax Print Center screen, select 2021 federal returns and Include government and TurboTax worksheets (optional) then select View or print forms.
6. The information will be on the first page, Electronic Filing Instructions for your 2021 Federal or State Tax Return next to Balance Due/Refund.

In the future may I suggest you pay the federal (and state if allowed) directly on the fed or state websites ... this way you have the control on the timing and get an instant confirmation of the payment.   https://www.irs.gov/payments 

Just curious why do you need another option rather than checking your bank account? I set up a bank account for my website to do that. That's seem pretty like the most straightforward way.

This totally answered my need