Hi Rob,

you seem to have a nerve entrapment which may be due to TOS.
I would guess that a strengthening exercise routine would make your symptoms worse even if you feel better after the exercise.
This is how it is for me. It took me close to a year (of daily pain) to realize that strengthening exercises actually contribute to the problem even though they seem to relieve the symptoms (probably due to improved blood circulation during the exercise).

Since laying down on the floor relieves the pain, this is probably what you should stick with for a few months. Do it several times a day combined with diaphragmatic breathing (to help relax the scalenes) and light stretching.
I found that if I give my nerves enough time to relax each day, the pain gradually goes away. I was able to stop all pain meds in a couple of months.
I am yet to figure out the long-term plan

As for the diagnosis, I think you need to rule out the spine problem.
If this is a soft tissue nerve compression then no MRI or EMG will show anything.