Jim wrote:

I understand if an open container of water is taken up to altitude
high enough, that it will boil away.

I take that to indicate when water is brought to a boil on our stove,
at sea level, that the heat actually activates or motivates the water
molecules to begin going crazy and rubbing against each other, and
creating the vapor which must escape.

More or less. The temperature indicates the average kinetic energy of
the water molecules. But in order for the phase to change from liquid
to vapour, additional heat is required. This is evidenced by the fact
that as the pot is heated the water in it will reach 100°C quite a
while before enough heat has been added to bring it to a full boil.

In any open container of liquid there is an equilibrium between the
rate at which molecules in the liquid phase evaporate from the
surface and that at which vapour from the air condenses back into the
liquid. Even at room temperature, if the air is dry enough, and
especially if there's a wind to carry vapour away, a glass of water
will evaporate. OTOH the water in a sealed pressure-cooker doesn't
boil easily because of the high pressure inside, and can be brought
to a temperature much higher than its normal boiling point while
remaining in the liquid state.

At any given temperature water -- and even ice -- has a
characteristic "vapour pressure"; as long as the partial pressure of
water vapour above the liquid is lower than the vapour pressure, more
molecules will be lost to evaporation than are gained by
condensation, and _vice versa_. We say a liquid is at its "boiling
point" when its vapour pressure equals the atmospheric pressure.

But if one were to take that open, cold pot of watet to a given point
above the earth's atmosphere, would the water get hot, as it would on
the stove, while it boils away?

No. By taking the pot high into the atmosphere you're lowering the
pressure; where this is lower than the vapour pressure appropriate to
the water's temperature, boiling will take place. Since the escaping
vapour carries heat away with it, the temperature of the water
remaining in the pot will actually drop; the same principle underlies
the cooling power of any refrigerator or air conditioner, except that
water (which is extraordinarily resistant to vaporization for its
molecular weight) is much less suitable for the purpose than Freon
and other such substances.

--
Odysseus