I own a second home in NJ which I have given to my relatives to live in and don't charge them any rent. What deductions can I take on my second home?

a) Can I take depreciation for 2nd home? How do I do this? Which form/method of depreciation to use?

b) Can I take property tax credit for both the houses? For NJ return, it only allows a max of $10,000 as property tax deduction. The property taxes that I pay for each house is over $10,000.

No you cannot depreciate a house that you are allowing your relatives to live in without charging them any rent.

Yes, you can still deduct property taxes on your Federal Schedule A. No opinion on NJ tax return.