Your list of symptoms makes me think of coronary vasospasm which is also known as variant angina or prinzmetal. It is basically a temporary constriction of the arteries.
I was diagnosed with this having had a heart attack at 46 out of the blue.
Of your list
Rapid onset
Severe pain across back and around sides of rib cage leading to
Shallow breathing
Crying with the pain
Pain/sensation going up into my jaw
Strange taste in mouth
Clammy and sweating
all could be heart symptoms
Lasts about 15/20 minutes
Not connected to stress or exercise
Variable frequency – can go months between episodes or have several in quick succession
are all seen in vasospasm which is not caused by exertion.
The problem with vasospasm is that it is very difficult to diagnose. If you have heart tests, they will no show anything wrong unless a vasospasm occurs at the time which is unlikely.
When I had my heart attack it took the cardiologists a long time to work out what had happened, and in the end they could only say - we think it was vasospasm (because no heart disease risk factors, no atherosclerosis etc) but we can't be 100% sure.
I have made a full recovery, just take medication (statin, aspirin) to reduce risk of artery disease and another medicine to keep my arteries relaxed. Also I keep a GTN spray in case of another attack - but four years on have had no further symptoms.
Sorry for long post. I wanted to give you this information because it might be worth you asking your doctor about this possibility - and it would fit with why tests so far have shown nothing. Also it is uncommon, most health professionals I meet know little or nothing about it and it is not usually at the front of their minds as an explanation for heart symptoms.
Just so as not to panic you, if it is this, what you would have been getting would be just angina (i.e. heart pain), just not caused by the usual factors, and not a heart attack, but you should get it looked into to reduce the risk of a heart attack in the future.
I hope that helps.