Would he have joined Voldemort?
It's possible that he might have still joined Voldemort. Dumbledore may have asked him to play the part of a double agent as he did in the books, although he probably wouldn't have been very convincing with Lily as his love.

Would his cruelty and bullying still have been there?
Cruelty and bullying might possibly have stuck around, although not as extreme thanks to Lily. He was bullied and mistreated as a child, and that would probably result in his bullying others when he's older. I imagine he'd be nicer to some extent, though.

Snape as a father?
Something about the way he was portrayed in the books tells me that he and Lily would never have children. But then again, that's because he was such a bully. Maybe if he and Lily did have children, he would pretty close to what Harry is like in terms of behavior and morals.

Would he taken more care in his appearance?
Even when Lily was alive and the two were close friends, he didn't care much about his appearance. So if he and Lily did end up together, I can't see any reason why he would take more care in his appearance, seeing as he's already gotten his dream girl. Although, there's a chance he would be more careful at Lily's urging.

Would he sing, would he dance?
I think of him to be sort of reluctant to do such things, but Lily is the opposite. I can imagine her trying to persuade Snape to sing and dance with her and succeeding in making him join in singing off-key and stepping on her feet with a red face from laughing and protesting.