Would a Professor be responsive to a meeting request from a PhD student like this?
@Penguin_Knight's comment is the best advice in this situation: ask Professor C to introduce you (via email with you in CC) to Professor X and mention that you will be in town and would like to chat.  Professor X may not have time to read all the messages she gets from students and other people she doesn't know, so there is always a chance your message -- not matter how well-crafted -- may be overlooked.  But a message from a coauthor will immediately stand out and is sure to be read carefully.

This is a very natural thing for Professor C to do (assuming she has a positive opinion of you and your work); you should make the request politely but you don't need to feel that it is a huge favor.

If you leave a good impression, this introduction and the subsequent visit will usually drastically improve your chances of admission.