I think the unexpected position, rotation, and scaling when exporting to .babylon can be fixed by referring to Mixamo - Babylon.js Documentation. Search the phrase Apply on this linked page (Position, rotation xyz all need to be set to 0, and scaling xyz need to be set to 1).
Edit: Sketchfab allows you to download in .fbx format (as well as .gltf). I’d recommend using the .fbx format. Then you can follow the guide linked above, and you should get the result files below. You can try dragging the .babylon tree in the .zip file below in https://sandbox.babylonjs.com/, and you should see your tree animating.
tree.zip (158.4 KB)