Answer 1

Since you're working with web languages, you may be interested in p2.js. It is a 2D rigid body physics engine written in JavaScript. Includes collision detection, contacts, friction, restitution, motors, springs, advanced constraints and various shape types.
You can find demo, Examples, Documentation, etc here
https://github.com/schteppe/p2.js

here's a direct link to the rag doll example using p2.js
https://schteppe.github.io/p2.js/demos/ragdoll.html

Answer 2

Custom in what sense?
Any specific programming language?

I would probably approach it this way:
Find implementations of ragdolls and study its physics and the code.
Clarify what my animation should do.
Clarify if I can use the code samples.
Decide if I can modify the existing code according to my ideas or if I want to try to write up my own version from bottom up.

Answer 3

Didn’t you just make this

https://sololearn.com/compiler-playground/W8tqOG551cNi/?ref=app
What else do you want? It looks pretty good as is!