Possibly not. That's mostly based on genetics. But if Lily did have Harry he would not look or have some of his personality. He gets most of his looks from is dad (except his eyes), James Potter, and personality has a bit of James in it. If Snape was Harry's dad, he would have greasy, black, limp hair and maybe cold, black eyes. Unlike Harry's real ones, witch are green, like Lily's.
And yes. Both Lily and Harry's last name would be "Snape".