You'll need to add your models as prop_static and enable per-vertex static prop lighting when buliding your map.

Unfortunately rotation_euler is only accurate if the bone is configured to use euler angles. It could also store its rotation as a quaternion (the default option) or an axis angle. As soon as you start converting them to eulers you're back to having rounding errors. In any case, it's not worth trying to handle all of those cases for the obscure requirement of making animation values match exactly.

Instead of messing with the exporter, I would suggest a post-process script which snaps values which are very close to the originals. That would work 100% of the time.

They aren't fixable. Floating point rounding errors are a part of life!

The message means that when weight painting, each vertex can only be attached to one bone.

It's impossible to say without your blend file.

The phrase "failed to" doesn't appear anywhere in the source code, which strongly implies that this error is being raised by Blender itself. It could be related to the corrupt character at the start of the file path? What is the square meant to be?

Mesh Properties > Shape Keys > the pin icon.

That sounds like the bad stuff. Replace each corrupt block of text with something else.

It looks like the file contains text which is neither ASCII nor Unicode. It'll be the name of a bone or material. You could try opening the file in a text editor and removing anything that appears corrupted.

Ask whoever created the importer you are using.