What repair options exist for iPhone 5s
If you bought the phone from an official Apple retailer, and it is new and within the warranty period, then there should be no issues with returning the phone for a refund or replacement.

The sales policies of Apple are at http://www.apple.com/legal/ and http://www.apple.com/legal/sales-support/sales-policies/retail.html

What complicates matters in many cases is the carrier. Unless you purchased an unsubsidized phone, you may have the carrier balk at a return and have to wait for them to clear your subsidy.

In the meantime, the issues sounds like the phone connector jack internal to the phone, you could attempt a DIY repair by inserting the headphone connector and giving it a good wiggle and spin about in the socket, this could get it working. If not, then read up on the policies and see if the store is willing to work with you to find a solution.