You can claim property tax deductions/credits if you file separate returns. As awesome tax Expert @marilynjoy said, you would have to take a 33%/67% split.
The New Jersey tax forms are set up to give you the better benefit between the property tax credit and deduction. Often, because of the credit for tax paid to NY, the credit works out better because the other state tax credit reduces most of your tax liability.
The property tax credit/deduction is separate from any standard deduction. New Jersey does not have a standard deduction so you can claim either standard or itemized on your federal return and it will not affect NJ.
If you file separate, the IRS will expect bank interest to be reported by the person who's social security number is on 1099-INT. You can issue your wife a 1099-INT and make an adjustment.
On a separate return medical expenses should be claimed by whoever paid for the expense. You can only deduct medical expenses that were not reimbursed or paid by the HSA.
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