1) push as much as you wish, so long as she doesn't resist. Her comment on the other guy might be an indirect way of saying "what are you waiting to start having sex?". Or maybe not, don't be pushy, you have to be a bit subtle.

2) it does not reflect bad on you. Your stories are different, and nothing says she's not investing more in you. If you want to get serious, I think you don't mind waiting a bit. But if it's a one night stand, that would be awkward. Also, if you feel uncomfortable, you might want to discuss with her about her views on sexual promiscuity, and state that you would prefer that she devotes herself to you. That's also a way of moving the relationship forward.

3) It does not matter who says it first. What matters is that you should only say it when you are pretty sure that she's interested in a relationship with you. Otherwise you may put an unwelcome pressure on her.