I note two seemingly important features in your profile.One is that you were diagnosed with 'Essential Tremor Syndrome' (ET) in 2001 and that your father has parkinson's disease.
'Essential Tremor Syndrome' is a progressive neurologic disorder mainly affecting the upper limbs, but may involve other areas as well- Characterized by intension tremors(kinetic tremors).
Some non-motor features have also been seen to occur with ET such as, depressive & anxiety symptoms as well as cognitive difficulties.
A familal pattern has been seen in some of the cases.About 50% of the cases may have a genetic mutation( presumably autosomal dominant transmission).
ET patients have a higher risk of developing Parkinson's disease.
There is no cause that i know of which can produce changes in body temperature on one half of the body. In your case, whether this represents a non-motor feature of ET needs to be looked into.
Please consult a neurologist who will be able to sort this out. Best wishes!
MB BS, MS, FRCS(Edinburgh)