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*I first spend between 30 – 45 minutes of the recumbent bike.
This is good. Keeping your aerobic fitness up is really important.
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*I do weights on the machines (very low weights, I have built up very little strength. I saw a personal trainer who helped me figure out the proper use and advised me to stay within 70% of my range of motion -or I seem to hyperextend certain movements. I don’t feel I’ve gotten any stronger over the last year and have only increased weight by 5%. Every time I try I pull a muscle.)
Weights should be good in theory. The most likely reason you are not building up strength is that the weight intensity is too low. If you want to strengthen things you have to be in the range of 8-12 reps where on the last rep you really can't do much more. HOWEVER if you are having so many problems this may not be feasible and I would be guided by your physio and personal trainer. Have you discussed your lack of progress with your weights?
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*Core exercises (my physio has sent me to a pilates school, where we do very careful core work, so I spend about 10 minutes doing some core exercises.
This is good! Should be helpful
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*stretches (basic for all muscles).
So are you doing these at the end of your exercise session? You probably should be going very easy on them just relaxing into the stretch and maintaining the stretch for 30 sec or so. You probably don't want to be over stretching.
As an afterthought, given that you have generalised hypermobility there is a group of conditions called Ehlers-Danlos Syndrome. I am not saying you have this but as you are having so many problems this might be worth getting checked out medically.
Also: do have a think about the Feldenkrais/Alexander idea