5. I paid the premium one week late (or one month late). Is my policy still valid? Will the insurance company deny my claim if an accident happened immediately before my premium payment?

There should be a "grace period" clause, which is another important contractual provision in the insurance policy.

Typically the "grace period" provides for a grace of 30 or 31 calendar days (depending on the terms actually written on the policy) to pay an overdue premium. The policy remains in force during the grace period. Thus, if a claim were made within the grace period, the insurance company is liable to pay the claim but the company has the right to deduct the amount of the overdue premium from the proceeds of the payment of the claim.

In the example above, the insurance company would be obligated to pay your claim.