Here is the link to a Landlord Tenant of South Carolina Pamphlet.
To answer your first question, No they don't have to renew at your original rate. You should try to negotiate with them to give you the same rate or not as much as they are asking for. It is typical to try to get a rent increase, after all the management company is trying to get the homeowner the highest & best rate. Most likely they are getting a fee based on that amount per month. On the other hand you are currently in the property & if you have a good rental history then you are in a good position to negotiate. If you moved out they would have to market the property to find another tenant. You may want to consult an Attorney specializing in Landlord Tenant Laws if you feel that you are being taken advantage of. Best wishes..
Century 21 Broadhurst & Associates