1. If you never physically worked in NY, you don't owe NY taxes. If you worked in NY for even 1 day (such as for training or required meetings) you may owe some NY taxes for those days.
1a. If you did physically work in NY for at least one day, then we have to apply the "convenience of the employer" test to determine if you all your NY employer wages are taxable to NY, or only proportionally for the days you worked in NY. It sounds like you would pass the convenience of the employer test and would not owe NY taxes, but this might need to be reviewed.
2. You would never owe NY taxes for income earned from a KY employer if you were not living in NY at the time. (Your wages before May.)
Generally, you file a resident state tax return for the state of your permanent residence that reports and pays tax on all your world-wide income. If you have income that is "sourced" or connected to another state, you file a non-resident tax return in that state to report and pay tax only on income sourced to that state. For example, if you won a NY lottery prize or sold property in NY, that would be NY-connected income. Most of the time, the "source" of wage income is the state where you physically performed the work, not the state where the company is located. Assuming you pass the convenience of the employer test, then your only NY-source income would be any wages paid for days you were physically working in NY (such as meetings or training).
Because you had NY withholding, you need to file a NY non-resident tax return to get it back. When you prepare a non-resident return, Turbotax will ask you to allocate your income to your home state and your non-resident state. In your situation, you would tell Turbotax that you had zero NY income. That will result in no tax owed and a full refund of the withholding.
