Blood TSH levels are 0.6 range is 0.5 - 5.5. I have anxiety depression and tremors and fast heartrate, is this related?

Possible. Tsh is "technically" in the low normal range so you cannot say that the thyroid is causing your symptoms at this point. It's still possible that the thyroid is the culprit but the only way to know is to check a complete thyroid profile: free T4 and free T3 (liothyronine) and possibly thyroid antibodies. The possibility is higher if the ft4 and ft3 is also elevated.
Possibly - see a doc. A stable TSH of 0.6 is very unlikely to be associated with symptomatic overactive thyroid. However, if the thyroid status is not stable, changes in TSH and symptoms may be out of sync by several weeks. It might be a good idea to recheck the TSH in a few weeks to see if is changing. Other causes of symptoms might be panic attacks, or too much nicotine, caffeine, or adrenaline, some drugs.
Low TSH level/tachy. Fast heartbeat with tremors and low TSH of 0.6, strongly suggest you may have graves disease(hyperthyroidism) you should discuss with your doctor or consult an endocrinologist.
Tsh. Wondering if you are taking any medications that might cause such symptoms.
Tsh. Tsh in low normal range could indicate slightly elevated thyroid function. Talk to your doctor about your symptoms and meds.
Not related. Tsh levels at the low end of normal are still normal and are very rarely associated with any symptom. You may want to repeat the test in a few months to rule out any progression.