After doing more research, I think I finally came up with the answer to my own question. Posting here in case anyone else has the same question(s).
Regarding this:
(1) What if part of my contributions were not tax deductible?
(a) On the NY TurboTax return, under Adjustments to Federal Income -> New York 529 college savings ccount activity, it asks for any "529 Withdrawals Not Used for Education". As instructed, enter the total of all nonqualified withdrawals, including previous and the current year. This feeds into New York 529 College Savings Program Worksheet , Part II, Line 1.
(b) It then says Enter Total New York 529 Contributions, Again, as instructed, enter the total of all contributions. This feeds into the same worksheet, Part II, Line 2.
(c) Then it says "Enter your nonqualified withdrawals and contributions from your 1998 through 2023 New York returns. However, what it is really asking for regarding the contributions is, enter the tax-deductible contributions made. In other words, if in 2010, my wife and I contributed $15,000, we should report only $10,000 for that year (because that was the maximum amount that we got a deduction for). This feeds into the same worksheet, Part II, Line 3.
Now, on that worksheet, Line 4 results in the total excess contribution beyond what was tax deductible. Line 5 adds in non-qualified withdrawals for previous years, resulting in Line 6. This amount is subtracted from Line 1. So you get the total non-qualified withdrawals, minus the non-qualified withdrawals taken in previous years (resulting the non-qualified withdrawal amount for the current year), minus the excess contributions. If you have more excess contributions than you have used, this amount is negative, and you report $0 additions on your NY return. If this amount is positive, this is the amount of non-qualified withdrawals taken out for the current year beyond the excess contributions. You report this amount on IT-201 Line 22.
So the bottom line here is that New York allows you take back any contributions made without receiving a tax benefit as a non-qualified withdrawal without having to pay back a tax deduction. It is only after you have used up your non-deductible contributions that NY will reclaim the tax benefit that you received when making the contributions.
For the second question, I tried this in TurboTax using the above logic. I "pretended" to withdraw the entire value of the 529 plan as a non-qualified withdrawal. Going through the above math, TurboTax subtracted the amount of "excess" contributions from the total, and then I reported the balance on my IT-201. So the answer to the question is that for non-qualified withdrawals, you pay tax on everything except for the amount of original contributions that were not tax-deductible. That is, you pay tax both on your original contributions that were tax-deductible, the earnings from those contributions, and the earnings from the contributions that were not tax-deductible.
(3) I still do not know how to handle the situation if an account is inherited, as this does not seem to be defined anywhere.
