This is not complicated at all. Ask her out and let her choose. Have a polite, but not too close relationship to the guy who is basically your subordinate. If you think he might harm himself, take action, but other than that your not responsible for his wellbeing.
Also, it sounds to me that you are trying to be absolutely fair to this guy, not only in this thread but also at work, but not necessarily like him too much. I’m also pretty certain that she was just polite to him and didn’t want to upset him. Talk with your love interest, than you will know.
It’s indeed important to be an understanding boss. And I’ve also been on both sides of this and agree with you that it sucks pretty much.
However: this guy is a bad worker with a bad attitude and clearly states that he has no interest in his job. He does bad enough work to upset his teammates. You’re his boss. Don’t hang out with him. Maybe a beer after work, but even that stretches it. You wanna be his friend and hang out with him outside of work? Then you both should not work directly with each other. There is a time and place for delicacy and sometimes not.
Sorry, I’m a bit harsh! But I had problems with a co-worker myself. And I suffered actually years from constant drama because I was polite and considerate and forgiving. And it would have saved me from so much frustration and stress if I kept my distance in the beginning or just plainly told him to fuck off.