I want to know which an alternative that I could paid less state taxes.

1st Scenario -
NJ resident working in NY starting from Jun
I understand that I have to file NJ-1040 and IT-203 for NY. Do I have to pay ny city tax?

2nd Scenario -
NY resident - have a property in NJ rented out to others. What do I have to do for NJ tax? NY tax? Currently, the rental income is a loss.

3rd Scenario -
NY resident working in NJ for the past 5 months (less than 182 days)
Do I get all the NJ state tax withheld as a refund?

Option
If you can choose between NY and NJ as your resident which one you pick to max the saving of state tax? NY or NJ?

1st. On your NY return you will report income earned while you were in NY. You will also pay city tax. On your NJ return, you will report all the income you earned while living in NJ even income earned in NY. On NJ tax return, you will claim credit of taxes paid to NY.

2nd. If you don't live in NJ during a year, then normally you don't file NJ tax return.

3rd. Same as 1st except states have reversed.

In general you will pay less tax as an NJ resident than as a NY resident, especially if you have any non-wage income, as NJ's tax rates are lower than NY State's. Worst from a tax perspective is to be an NY City resident, as then you pay both NY state and NYC income tax.