If God gave people over to a reprobate mind as it says in Romans 1:28, does that mean he hardens their heart against him and his word?

No. Paul is describing humanity before it meets Jesus. This is almost a historical look into how the need for a Savior arose. Because humanity had become darkened in its mind and its thoughts were evil, God had to intervene.

The hardening of the heart (like Pharaoh) only happens after the person refuses to listen to God a number of times. When you read the account of Pharaoh pardoning his heart, he did this one or two times before God made it permanent and hardened his heart.

There is still hope for anyone who is not met and received Jesus. The hardening of the heart comes after multiple attempts by God to bring a person out of this state and condition of Romans 1:26.

The opportunity for those who have had the dark in the mind and reprobate state changes the moment they had the gospel and accept Christ as their Lord and Savior.