I have an HDHP + HSA and my spouse has PPO health insurance through our employers. Can I use my HSA to pay for his medical expenses, such as coinsurance and copays?

According to IRS Publication 969, you may use your HSA funds for qualified medical expenses for "you and your spouse", among others.  See the information from that publication, linked below.  Look specifically under "Distributions From an HSA", then to the sub-heading of, "Qualified medical expenses".

https://www.irs.gov/publications/p969/ar02.html#en_US_2015_publink1000204081