Short answer: Yes to both. The VA is required to bill private health insurance for care of non-service-connected conditions, and the veteran can use HSA money to pay VA copays for that care.
The VA bills the veteran’s private health insurance, including a spouse’s plan that covers the veteran, for care, supplies and medicine for non-service-connected conditions. It does not bill Medicare or Medicaid, but may bill a Medicare supplement. The veteran does not owe the unpaid balance the insurer will not cover, though a VA copay may apply depending on priority group. Insurer payments can offset VA copays, and the insurer may credit VA charges toward the deductible.
The VA may bill HDHPs and accept HRA reimbursements for non-service-connected care, and a veteran with an HSA can use it to pay VA copays. Having other insurance does not reduce VA eligibility.
HSA eligibility itself is a separate question: receiving VA medical benefits in the previous three months can block contributions, with exceptions for service-connected disability care and preventive care (see the existing VA and HSA card).