About how long do I have to wear my mouthguard for TMJ before my symptoms start to go away?

It varies. Depending on your individual situation. I always counsel patients that they need to wear the appliance for at least six months before we consider that the appliance may not be working. Make sure the appliance is custom made for your mouth by a dentist. Over the counter appliances do not work and can make problems worse.
Variable. Variable. Sometimes within a few minutes. Other times much longer. If you have no relief within a week or two, probably not working. See a TMJ expert to help evaluate and treat your problem. A TMJ expert can be general dentist, an oral surgeon, prosthodontist, orofacial pain specialist. Ask your physician, dentist or dental society for a recommendation.
May not without help. If just wearing a mouthguard is the only treatment you are depending upon for healing, you may not be headed in the right direction. Often, a mouthpiece is only one piece of the puzzle. Exercises, physical therapy and other treatments may be needed in addition to the mouthpiece to help you. Another type of mouthpiece may also be needed.
It depends . It depends on the severity of the condition and the cause and the symptoms.The occlusal guard can provide some immediate relief.It must reduce the activities of the muscles and reduce the response of the nervous system as well. In addition to the occlusal guard you may need some anti inflammatory and muscle relaxation medications. Talk to your dentist if the symptoms persists for more them a week.