"If I am able to get the HSA distribution treated as a mistaken distribution, would I still answer yes when TurboTax asks if I've taken any distributions, or could I answer no and skip that part?"
If the only distribution (1099-SA) that you received was for this mistaken distribution, leave it off your return, because it will confuse things when the HSA custodian issues a corrected 1099-SA.
"am I able to change the distribution code from 2: Excess Contribution to 1: Normal Distribution?"
As a general rule, do not change data on a form sent to you, because the IRS is sent a copy, and they may wonder why you changed the form.
"I called my HSA provider and it seems they will honor a request to treat it as a mistaken distribution. However, they said that the corrected forms will not be available until May of this year. "
This is good news.
"I don't understand how to go about reporting both the initial distribution and the return of funds as part of a mistaken distribution."
I assume that you told the HSA custodian that the initial withdrawal was for excess contributions. We now know that it wasn't (because it wasn't any excess for 2021), but I assume that the 1099-SA (have you received it yet?) will show a distribution code of "2". Go ahead and enter that as is. This will not make much difference to your tax return.
Personally, I would file an extension (form 4868) as soon as you can, because life for you will be easier if we do all the following in the original return and not have to file an amendment. Just because an amended return has a due date of October 15th, doesn't mean you have to wait that long.
However, if the distribution code is "1", then you will have to enter it as is and count on the corrected 1099-SA showing up in a timely manner.
In May, when the corrected 1099-SA arrives, enter it instead of the original 1099-SA. Be sure to check the little box on the second screen of entering the 1099-SA that says "The corrected box is checked on this 1099-SA".
"When next year (tax year 2022) rolls around, will I need to do anything to report the mistaken distribution that was returned?"
Don't do anything about this mistaken distribution.
"if there is an easy way to convert the distribution code from 2 to 5 and just pay the penalty and be done with, I'd consider it. Please let me know if this is possible."
As noted above, don't change data on forms when you enter the forms. This just invites letters from the IRS.
In any case, let me summarize.
1. File an extension. TurboTax supports this form, but you can't e-file it. Just print and mail the 4868 as quickly as possible.
2. Now you have some time to get other matters settled. Wait for the correct 1099-SA to come. When it arrives, enter it instead of the original 1099-SA (because it will confuse everyone). When the IRS sees a second 1099-SA with the correct box checked, they should know that the first one should be ignored (since we are not going to enter it anyway, that's OK).
3. For tax year 2020, two things:
A. TurboTax should know that you are carrying over an excess (yes, from 2019). So I don't think it will ask you if you overfunded in the previous year (because it already knows). If it does ask, be extra sure to review the 8889 so that you don't end up telling TurboTax that you have twice the carry over that you should.
B. BE SURE to reduce your normal HSA contributions by $1,500, so that this 2019 excess will be used up in tax year 2022. Too many taxpayers forget and max out the contributions and end you propagating the excess.
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