Does the "Pay with my Refund" option cause an additional delay in direct deposit, or does the deposit take place the same day as the "Expected Federal Refund Date"?

TurboTax does not receive any kind of tax refund, federal or state.

If you requested to have the TurboTax account fees deducted from your federal tax refund, the IRS will send the refund to a third party processor to have the fees and the service charge deducted. The third party processor will deduct the TurboTax fees plus the service charge and then attempt to direct deposit the remaining refund balance into the account that you entered on your tax return.

For the status of your tax refund you will need to contact the third party processor, Santa Barbara Tax Product Group, using their website - https://www.sbtpg.com/

Tax Products Group Contact Information - https://help.sbtpg.com/hc/en-us/articles/230084587-Contact-Information