I don't know how much help this will be but...
My allergy is no where near as bad as what your discribing. It rears its head when i use normal bandaids or if i wear gloves made of latex. From the bandaids, I get hives that last for about a week, and the gloves cause my hands to itch and turn red.
I have no issues with elastic in the underware or any other garment. You can, however, find undies without the elastic in them, as well as other clothes.
For bandaids i use the Nexcare brand. They mainly use latex free items. For the gloves, i have found a pair of lined rubber gloves, however, i am unsure of them as of right now.
From my understanding, different rubbers are used for different things. I have no issues with say, the rubber on a basketball.
Yes, I am having to be pretty careful with the elastic. Any rubber I am leery of...(You are right--there are 3 different "rubbers" that we can react to. Most common--elastic, rubber gloves, goggles & of course, my fav: the cell phone cover!
I have some very heavy duty rubber gloves that are lined too--then, I wear my little white 100% cotton gloves inside of them with the tops of the gloves over long sleeves. In an pinch, this has worked, and it has not caused me problems! I am guessing if you have similar high quality gloves with a lining--that you might be ok!!! (I hope) Thank you for the input--I will be looking for 100% cotton everythings!