I am aware that a landlord can refuse to prorate rent once someone has signed a contract to pay through a certain time, but can a landlord refuse to prorate at the beginning of a lease? I am moving into an apartment on September 10th, and the receptionist said something about prorating only beginning 2 weeks into the month. According to the law, will I have to pay for those first ten days, or can I negotiate my lease so that I begin paying on the day I move in?

Well, you can certainly ask for and attempt to negotiate anything you want. If they'll agree to it or not is another matter.

You can check your state's landlord / tenant laws in a sticky at the top of this forum. It may be addressed in the law how the proration has to be done.