For your "capsule" shapes, Maya has a capsule/pill primitive that you can use that will solve the "interpenetration" issue for you.
I'm not sure if it is a default primitive, or an "Unlimited-Only" primitive OR it may be a feature of the Bonus Tools download package.
If you don't already have it, you can get it from the Audodesk AREA community site, but it is free.
For your other issue of going from a rigged character to a "limp pin-constraigned"-style of dynamic passive-rigid ragdoll, the only way i can see that happening off the top of my head would be to switch between 2 versions of your character in post.
I'm not a character animator, so you may have to post that specific question to the animation forum
But anyhoo, as the fall starts, pick a frame you would like the switch to happen, pose the "ragdoll" character to match your falling rigged version, and simply animate thier visibility so a switch is made.
You can hide the switch in post with either motion blur, or a full-on cut to another camera.
Israel "Izzy" Long
Motion and Title Design for Broadcast-Film-DS