Picking Optimal Altitudes

When planning a flight against the wind, how to you pick the best
altitude when trying to minimize flight time?

If I read my performance charts correctly, my aircraft (Piper Turbo
Arrow III -- service ceiling 20,000 ft) seems to gain about 2 nts of
*true* airspeed for every 1000 feet you go up. Firstly, does that sound
about right to folks? I assume that this effect is from the decrease in
air friction at high altitudes (even though the prop also has less air
to "push on").

So, unless there is some other factor, I think this means that if the
winds increase at anything higher than 2 nts per 1000 feet, I am best
off staying at the Minimum Enroute Altitude. Is that right?

My experience thus far suggests that most of the time, the winds aloft
speed increase far faster then 2 nts per 1000 feet, so, in general, it
is unlikely that I will do much better than staying at the absolute
minimum altitude. Is that consistent with other folks' experience?

-Sami