Does the end of Romans 1 teach that homosexuals are given over to a reprobate mind and cannot be saved?

More or less, that is what Paul is saying. He says it three times in that context, that God gave them over to the lust of their desires, and when God does that it is like the final curtain, when he gives up on prompting you with his Spirit to repent, what we call actual grace. When you reach that point where God has given you over, there is no chance for your salvation, or if there is a chance it is as slim as a strand of wheat in a farmland a billion acres big. God is not a sappy, milk-toast figure who is there for you whenever you decide to repent; if he gives you the truth and you turn your nose up at it once, possibly twice, it is over. This is like Hebrews 6, which says that after you have tasted the good word of God and the gifts of the Spirit and then turn away, it is impossible to renew you again, using that exact Greek word for impossible. So triple that with Romans 1, where God says he has given them over to their own lusts, and the last verse says it comes with a punishment, meaning they are going to die in their sin. There you go.