Usually not. Interest on a carried balance is often much larger than the cash back, so pay the full balance each month if you want the rewards to help.

Some cards rotate categories or cap rewards. Check the terms, since the rate you see on a promotion may not last.

Only if the rewards you earn are more than the fee. Use the after-fee result to see whether it is.

Yes, at the same rate. But spending more to earn rewards is only a gain if you would have spent that money anyway.

Cash back is generally treated as a rebate, not income, but rules can differ for some rewards. Check current guidance or a tax professional for your situation.