do you buy new shoes?

You're question is a little confusing. If you mean do they sell new shoes, then the answer is yes. If you are asking if they will buy shoes from you, new or otherwise, the answer is no.

Yes

They have two sections, discounts in the back portion of the outlet, sizes and styles are what is on the floor. The front section is new styles, not necessarily discounted.

Yes at a discounted price espically in the clearance section.