"My primary issue is that I have a $0 limit for 2022 traditional IRA contributions"
That's different than I had thought. I had assumed that when you said that you had an excess traditional IRA contribution that you meant the $93 of gains that were transferred to the traditional IRA form the Roth IRA in the recharacterization and I interpreted the part where you said the initial (Roth IRA) contribution was entirely an excess contribution because your MAGI was too high. The only reason that you would have a $0 traditional IRA contribution limit is if you (nor your spouse if filing jointly) had any compensation to support a traditional IRA contribution.
If you truly had no supporting compensation, making you ineligible to make an IRA contribution of any kind, yes, you'll need to obtain a return of contribution. Because funds that were ineligible to be in the traditional IRA were converted to Roth, it's a failed conversion and the deposit into the Roth IRA ended up being a regular contribution that would have to be returned. Getting the IRA custodian to understand everything and process the return of contribution correctly will be a significant challenge, as will explaining everything to the IRS. Ultimately you'll have no traditional or Roth IRA contribution for 2022 and you might have some taxable gains to report.
1) No, the financial institution is not limited to doing a regular distribution, but they need to be made to understand that this was a failed conversion and a return of contribution from the Roth IRA can be made. If they will only make a regular distribution, you'll have an even bigger challenge reporting everything on your tax return.
2) I think that you'll need to treat the distribution from the traditional IRA that was "converted" to Roth, combined with the $93 distribution as having returned the entire original traditional IRA contribution, then treat the deposit into the Roth IRA as a separate regular contribution of $7,000. You would then request a return of that $7,000 Roth IRA contribution.
The alternative would be to report the $7,000 excess Roth IRA contribution for 2022, pay the 6$, $420 penalty for 2022, then obtain a regular distribution of exactly $7,000 (or, if eligible, apply the $7,000 toward your 2023 Roth IRA contribution) to eliminate the excess on your 2023 tax return.
