Mild forms of tmj may be self-treatable - some achiness, occassional ear aches or ear pressure, headaches etc. But when it becomes chronic or interferes with jaw function or normal daily life it needs to be treated professionally. It can progress and produce further damage if not treated. Surprisingly, at least to me, previous orthodontia treatment is fairly common among tmj sufferers. There are different theories as to what the correct bite is but many drs will agree that an open or severe overbite contributes or predisposes one to tmj later on.

Clenching and grinding can cause eventual tooth loss since it promotes bone loss (you mentioned you already have gingivitis, this also leads to bone loss if not treated so improve your oral hygeine routine if its not what it should be) however the sensitivity is probably more due to those teeth experiencing more pressure due to a poor bite and/or the affects of the chronic clenching and grinding - a proper spint or mouth guard would equalize the bite and alleviate some of that.

The cough is failry common as well - it has to do with the sinuses being inflamed. The clicking when you cough indicates your tm joint is being affected in some way - probably not properly positioned but sometimes even congested sinuses can cause this.

Jaw exercises may help ease some of the pain as an interim solution - I don't see how they can harm you assuming you're careful to not over extend your jaw but I doubt they will correct or resolve this problem.

There is a previous post you can search for that lists many self-help or home remedies that you can try but I think your best chance in resolving this is in obtaining a splint. Maybe you can check if your medical insurance would cover it or go to a dental school or university near you since they tend to charge less than drs in private practice.