You are correct in saying that LSS decreases as you climb. That is why at sea level, when (using the above figures) you travel at 650KIAS (@ sea level = 650 TAS), you are just below the speed of sound, where as when higher up, you might have a TAS of 500 and a M0.92 (92% of the LSS). Obviously multiplying 500 by 100/92 equals approx 540 TAS.
With regards to the effects of TAS...
True Air Speed is exactly that. Your airspeed through the air, so it does not vary with altitude or temperature.