The symmetry of the IK controllers is bad, here is what we have currently:
If I delete the right leg and symmetrize the left leg, here is what I have:
Now it works correctly, except I'm not sure about your feet Copy Rotation constraint but it's another problem... (I would rather parent the feet to the controllers and give them a Copy Location with the ...
Click on the bone that you want to connect to the other, then shift-click the bone you want to connect to, then press P to parent. A menu will appear and just click on "connected". This will connect the bones and parent the horizontal bone to the vertical bone (I'm assuming this is what you want to do).
Maybe do it through the N panel, choose Euler instead of Quaternion and enter the value you want on the rotation axis you want.
I guess if you do it manually Blender will try to find the shortest path from position A to B, whereas if you ask to rotate 500° in the N panel it will do it for real.
I believe from looking that they are separate meshes, in which case you must join them with Ctrl-J and then you should be able to weight paint it, if not you could separate the eyes and parent the separated object to a bone so that it rotates as if it were attached (but this would not morph the mesh like regular bones would).
glTF has a straightforward node hierarchy, where children always inherit all of their parent's transforms. It may not be able to exactly match Blender's options for things like "Inherit Rotation" specifically.
That said, there is an "Always Sample Animations" checkbox under the animation section of the glTF exporter. This is on by ...