Thank you for providing all of the necessary assets. It seems that the main issue may be how your skeleton hierarchy is set up for your car. Having the door’s direct parent being the root would explain why it falls off as soon as you begin to simulate. You may want to look at the Advanced Vehicle template to see the skeleton used there. You also have the option of using sockets for these pieces instead of having them be part of the actual mesh.
As for the door breaking off, it would be best to set up a breakable constraint for the hinges. These settings can be found in the expanded settings under Angular/Linear limits in the Constraint Mode of PHaT. A good place to learn about that would be the Physics map of the Content Examples project.
I hope this is able to point you in the right direction!