Heather: You need to find out what flame retardants are used. Some flame retardants are designed for use in thermoplastics and can withstand temperatures well above 220F. Other flame retardants may be extracted by steam cleaning. Extraction would put the flame retardants into your waste water and reduce the flame resistance of the carpet. You may also find that the flame retardants used in polyurethane foam (such as you probably have in your sofa) are very different than those used in your carpet.
I suggest you contact the manufacturers of these items and ask them what flame retardants were used and their cleaning recommendations.