The act of paying someone a compliment is fine. The thoughts and intents of the person doing this are what make all the difference, though. If one does it with the hope or fantasy that the other person might respond in a way that could potentially and eventually lead to overtly sinful conduct, then one has sinned in one's heart. Some folks might flirt as a way to stroke the other person's ego, for the desired purpose of causing the other person to stroke the flirter's ego in return. This is selfishness, which is also wrong. I hope that is helpful.
Maybe. It can be a way that someone can indicate romantic interest in another person leading to dating and marriage. However, as Rexion has said, there may be other motives which could make it sinful.
I must confess, I was thinking of flirting from the standpoint of a married person flirting with someone other than the spouse. For the unmarried, I think some flirting often is necessary to make the other party aware of one's interest in exploring a possible "get to know each other" relationship. Done with the right heart motive, there's nothing wrong with that.
Flirting in and of itself is not sinful. The reason behind it can be or it can be harmless and fun, or it could be a way to signal interest in someone but the flirting is not sinful but the intentions behind it can be.
Does it feel like a sin? If so, the intentions are probably not the best. Otherwise, this is one of those questions that Roman Catholics always seem to post, asking what is a sin and what isn't and worrying about sin all the time. No one on an internet forum can tell you if you have sinned in your heart or not - work this out in your relationship with God, and if necessary, speak with your priest, but just don't get all full of scruples worrying about sin all the time. Much better to focus on the love and mercy of Jesus Christ instead.