1. No, the credit for taxes paid to another state in general applies if you work and pay taxes to a nonresident state, then have to report the same income on your resident state tax return. Because you were a part-year resident of Illinois and New York and worked remotely in the state while you were a resident, you only have to report the income to each state based on when you earned it.
2. The October to December income would be New York income because you earned it while you were working in New York. New York source income would be if you worked there as a nonresident, or if you had a rental property located in New York.
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