Can a high-normal dose of levothyroxine cause bulging eyes and skin around eyes to swell? or are they only caused by natural hyperthyroidism?

With hyperthyroidism: What you're describing is called "Grave's ophthalmopathy", with the eyelid swelling causing eyelid retraction with a bulging eye appearance. It almost happens in hyperthyroidism from an autoimmune disorder (Esp in Grave's disease) . If you're taking levothyroxine and have your thyroid function monitored, then you shouldn't worry about this problem.