Are the limbs of the 3D model rigged to the mesh? Or are you using BoneAttachment3D nodes?
Looking at the first picture provided, there doesn't appear to be a Skeleton node in the scene, so it would make sense that it is not moving when the skeleton is animating.
Also, if the Skeleton and the mesh are not in the same 3D file, then you'll need to somehow bind the meshes of the character to the Skeleton, as otherwise they will not move when the skeleton moves because the vertices of the mesh are not rigged/assigned to any of the bones in the Skeleton. You may want to rig the character in something like Blender, as I do not believe Godot has any built-in rigging support or plugins made for this purpose.
However, if each limb is a separate MeshInstance node, you can use a series of BoneAttachment3D nodes attached to the Skeleton, one for each bone. Then you can have each limb's MeshInstance node as a child of the BoneAttachment3D node bound to the bone that the mesh represents. Then when the Skeleton is animated, the BoneAttachment3D nodes will follow along and because the MeshInstance nodes are children of the BoneAttachment3D, they will follow along too.
As for the code to play animations, you are correct that it is something like get_node("AnimationPlayer").play("AnimationNameHere") or$AnimationPlayer.play("AnimationNameHere")`.
