Just social, probly. But it's still an opportunity.
not really, its normally done as a polite gesture to someone. Like how you leave best wishes for someone on an email even though you might not, its just polite. Maybe she's close to you but writing love whatever her name is, on a birthday card doesn't really mean that much, certainty not that she is in love with you. there would have to be other signs
She sure doesn't hate you, try to break the ice some more and find out.