Very simple answer to your question - however, you should always confirm policy details with your credit card's insurance carrier, and your own private insurance company.

1. using a free day coupon or award, or a collection of coupons that results in $0 of rental charges on the contract (only taxes) - no credit card coverage whatsoever

2. using a free day coupon or award, or a collection of coupons that still results in one or more days of actual rental charges on the contract (ie, rent 5 days with 2 free days) - you should have credit card coverage

3. your private insurance does not care about coupons/free days

4. if you use a corporate code with LDW, then all credit card coverage is voided because the policy requires you to decline rental car company provided coverage, however in almost all cases, your personal insurance will cover damage and liability