When we have thyroid disease the quality of our tears change. They become more watery, and they do not lubricate our eyes as well. What you are calling "eye fatigue" when you are looking at a computer screen, etc., could be, in fact, dry eyes. We have to be extremely careful if this is going on, and put artificial tears in our eyes to prevent corneal damage. The drops we need are not the "get the red out" type of drops, nor are they the drops for allergy issues with the eyes. The products are labelled "artificial tears."
Double vision is a part of Graves’/Thyroid Eye Disease. Whether or not that is why you are experiencing double vision occasionally is another question, and one only the opthamologist can answer.
Lid problems may be a result of the thyroid imbalance.