Zathura:

1) You must file a tax return to settle your taxes. If you check several other postings, you will be able to review the three options available to you:
- File as a non-resident alien (for 1040NR)
- File as a dual status alien (Form 1040 with 1040NR attached as a statement).
- File as a resident alien (Form 1040) sometime in March.

Regardless of which way you choose, you can claim the standard deduction of $5,000.

2 & 3) Because you worked and lived only in Tennessee, you owe zero NJ tax. Instruct your employer to stop withholding NJ tax. File a NJ tax return as a non-resident to claim a refund of all taxes withheld. Since TN has no income tax, no state taxes should be withheld henceforth.