vasodilation promotes heat loss to environment.
Vasodilation does promote heat loss to environment except in the case when the heat outside the body is greater than inside. At which time, the process reverses itself and the heat travels down the concentration gradient into the body.
nerve impulses in sypathetci nerves leading to adrenal medulla slow(or stop) the release of epi and norepi, wich DECREASES METABOLISM.
Correct and I agree.
muscle tone decreases, and therefore heat production
(this is the opposite, or the lack of, shivering)
Again, I agree completely.
thyroid gland produces less thyroid homone, again LOWERING METABOLIC RATE.
I'm not sure this is an immediate affect. However, I do agree that whenever it does come into play that it certainly does as you say.
I believe all of this would prove the BMR to be lowered, except as I pointed out above, it would seem to me the outside (the body) heat would keep the body's BMR rate at about the same thus countering the negating effects from the body's feedback systmes.