Can I make one state tax-free by adding a 0% rate for it?

No. A 0% rate doesn't cancel other rates. At checkout every rate that matches the address is added together, and 0% rates are skipped because they add nothing. Your store-wide *All ()** rate still matches that state, so it is still charged.

To leave one state untaxed:

1. In Settings → Store Settings → Manage Tax, click Delete on the store-wide default rate (marked Default in the table).
2. Add a rate for each state you do tax, with that State selected.
3. Add nothing for the tax-free state.

Customers whose address matches no rate pay no tax. Check with your accountant that this matches your obligations.