For anyone who has similar situation. You also need to file Schedule R/NR besides Form 1-NR/PY according to this page about residency status. Schedule R/NR says:
"If you received Massachusetts source income before or after you became a Massachusetts resident for part of the year and your gross income exceeded $8,000 for the year, you must file a Form 1-NR/PY as both a nonresident and part-year resident."
So my assumption in Question 1(part-year resident does not pay tax after residency ended) was incorrect.
