I thought I heard (though I can't remember where I heard it) that homelessness is mostly a matter of mental illness and drug addiction more than poverty.

This dovetails with everything I know about homelessness. What's worse is that I see super young kids taking to panhandling. Maybe at that point in their lives they're not suffering from mental illness per se (not going to get into what's lacking at home that compels a young kid to panhandle) but the life of homelessness is so rough that it surely precipitates terrible consequences for mental health, physical health, esteem, direction, etc. But homelessness is a distinct though tangentially related issue to vacant housing. They both indict a broken system, broken in ways that people argue about to this day.

Also, wouldn't giving away empty houses hurt the real estate market even more, leading to more unsellable, and soon-to-be-empty-in-the-future houses?

I'm not sure by which mechanism you refer to the real estate market being hurt by. Again, I don't think the solution to homelessness is giving away keys to empty houses (many of which in Baltimore are condemnable). Do you mean the real estate market would be hurt because no one wants to live near mentally ill people? Because mentally ill people don't take care of their property and so values go down?