I think the girl should be approachable as well as able to start or continue a conversation with a guy. I do not think the guy needs to perform like a peacock to get a girl's attention though if a girl is very interesting to a guy, they will do things to catch her attention.
I believe the guy should ask for the date and it should be a date, not an offer to "hangout" or some other vague thing that leaves the girl wondering if the guy wants romance or friendship or is just trying to convert her into booty call.
I think girls should be polite and if she does not want to date a guy, she should politely decline, not accept then cancel or give a million excuses that keeps a guy wondering if he still has a chance when you know you are not interested.
So, no, I do not think girls should chase guys but they also should not leave everything up to the guy either. Dating and romance is a dance between two people and both sides need to bring something to the table in most cases.