Water in vacuum (or space) and temperature in space 
*

*So, water in vacuum will boil first and then freeze. I don't know how the freeze happens. As pressure lowers to zero, what happened to freezing point? (I know heat taken by vapor, and the water cool down, but I don't think it will be that cold, will it? In vacuum, boiling point is so low that water shouldn't need so much heat as it does in normal pressure, which means vapor actually takes more heat away under normal pressure than in vacuum, so water under normal pressure would be cooler? (I'm guessing) 

*And temperature comes from heat generates by motion of molecules (I guess so), so in vacuum, there is no temp? 

*What happen when I heat up a vacuum tube? 

*Does heat need a medium to “travel"?
 A: Conventionally, though with justifications, space is said to begin at the Kármán line which is
100km from Earth surface, i.e., still pretty close. The atmospheric
pressure at this altitude drops to about 0.032 Pa (wikipedia), which is still a
lot more than outer space (less than $10^{-4}$ Pa according to wikipedia)
The phase diagram of water shows that, at this pressure level, water
can exist only as a solid or as vapor, depending on temperature, but
not as a liquid. The phase transition between solid and gaz at that
low pressure takes place near 200°K (around -73°C), which is not that cold.
So, if you drop in space a blob of water at room temperature and
pressure it will instanly start to evaporate (boil) and decompress.
Here I am not sure about what happens. There are accounts from
astronauts on the web that explain that the water (actually urine)
will first vaporise then desublimate into tiny crystals. But no
explanation of the actual physical phenomena that drive it.
My own reconstruction of what could happen (before I saw these sites)
is the following.
First the loss of pressure propagates very fast in the liquid (speed of
sound?) while loss of temperature (heat) propagates slowly (as all beer lovers
know from their fridge). So the boiling will essentially take place
uniformly in the whole liquid. Phase transition from liquid to gas
absorbs heat, and that is what will cool the water very quickly, as
it evaporates.
My guess is also that the energy loss will cool the water down to
sublimation temperature (solid-gas transition) before it all
evaporates, so that some parts of the liquid may be cooled down to
freezing before they have time to evaporate.  But as boiling takes
place everywhere, it actually breaks the remaining water into tiny
fragments that cristallize, and possibly also collect some of the
vapor to grow.
Anyway, you apparently get snow.
But the cooling is due to evaporation, which is very fast,
much more than to radiation which has hardly any time to take place.
Numerical evaluation
We analyze what becomes of available heat to understand whether some water freezes directly. This is a very rough approximation as the figures used are
actually somewhat variable with temperature, but I cannot find the
actual values for the extreme temperature and pressures being
considered.
The specific latent heat of evaporation of water is 2270 kJ/kg.  The
specific heat of water is 4.2 kJ/kgK Hence, evaporating 1 gram of
water can cool 2270/4.2 = 540 grams of water by 1°K, or 5.4 grams by
100°K which is about the difference between room temperature and water
(de)sublimation temperature in space. So my hypothesis that there is
not enough heat available to vaporize all the water is correct, as
only about one sixth of the water can be vaporized with the available
heat.
Out of 5.4g of water, 1g will evaporate, though may cool down to just
above the sublimation temperature of 200°K, while the remaining 4.4g
will be cooled to sublimation temperature without vaporizing,
yet. The remaining 4.4g cannot remain liquid, hence, one part freezes,
thus freeing some latent het for the other part to vaporize. The ratio
between the two part is inversely proportional to the specific latent
heat for freezing and vaporizing.
Latent heat for freezing is 334 kJ/kg.
The sum of both latent heat is 2270+334=2604 kJ/kg.  These figure are
very approximate. As a sanity check, the latent heat of sublimation of
water is approximately 2850kJ/kg (wikipedia), which  show that the
figures are probably correct within a 10% approximation.
The ratio divides the remaining 4.4g into approximately 3.8g that
freezes and 0.6g that evaporates, making it a total of 1.6g of
vaporized water.
So, skipping a quick calculation, we find that about 70% of the water freezes into some kind of snow, while the remaining 30% are vaporized. And it all happens rather quickly.
I was actually uneasy about this account of astronauts stories of
water boiling and then desublimating at once, because that would leave
us with all the heat to get rid of very quickly. How?  Does anyone have a better
account?
A last remark is that there always will be some part of the water that
gets frozen. I thought initially that very hot water might provide enough heat to vaporize itself completely un low pressure. The critical point of liquid water is at 650°K (with a much higher pressure than you care to create in space:  22MPa), which is
only 450° above the sublimation temperature. But the water should be
cooled by 540° to provide enough heat to evaporate completely.  So
the water temperature will drop to the sublimation threshold before
enough heat can be supplied to evaporate it completely. This problably
a very simplistic analysis, though. I leave the rest to specialists.
A: Heat transfer happens by three methods, convection, conduction and radiation. Only radiation happens in vacuum, because unlike the other two methods, it's the only method that doesn't need a material medium.
The temperature of the water doesn't drop on earth (sea level), because as the water radiates heat, it receives that heat back, by radiation falling on it from the surrounding matter including the air around us, putting it in a state of thermal equilibrium with its surroundings.
In vacuum, that heat lost due to radiation won't be replaced, therefore the water would lose that heat at a much higher rate and freeze.
Regarding your question: "so water under normal pressure would be cooler??", no, the only difference is that in vacuum, the temperature of the water is more than sufficient to boil the water due to the lack of pressure. 
At sea level however, it needs to be "hotter" ~(100 C) due to the higher pressure. It will use that heat to evapourate, get a bit cooler due to the phase change, and the vapor will then cool gradually to match the surrounding temperature by the three methods I stated above, Check this for more details on HT methods: Heat Transfer.
Regarding your second question; about heating a vacuum tube. What will happen is that the tube's material itself will heat up. However if another object, was placed somewhere inside that tube it will receive heat by radiation (the same method heat reaches us from the sun) from the inner wall of the tube, and will start to radiate heat (electromagnetic waves) itself.
That object will stop getting hotter when the heat it radiates is equal to the heat it receives from the inner wall of that tube.
Please note that you haven't provided any dimensions or quantities regarding the water or the tube so my answer towards both experiments is a general case.
A: Your explanation is correct. The cooling is because every mole of water that evaporates removes one molar latent heat of vaporisation.
The latent heat of vaporisation is not pressure dependant, or at least is only very slightly pressure dependant, so the evaporation cools the water to zero Centigrade then freezes it.
A: It seems that most places I've read (on the web) people refer to the depressurization of water (in a vacuum or space) as "boiling" but I have very rarely seen this referred to as an out-gassing of the water's internal dissolved gases (nitrogen, oxygen, CO2, or any other gases that may be used in a space vehicle). However, unlike traditional boiling which releases vapor molecules (H2O) from the water, placing water in a vacuum chamber does cause the water to appear to boil but upon repetition of this experiment (re-exposing the same water to the vacuum) the "boiling" event is significantly reduced, yielding the result of freezing without much disruption of the water. It has been several years since I witnessed this experiment repeated in a lab vacuum bell, but the best explanation at the time was that the quantity of dissolved gases was much lower in the water sample when the same sample was relatively quickly re-exposed to a vacuum. What I don't recall is whether the water took longer to freeze upon subsequent exposure to vacuum or not. If the subsequent exposure to vacuum took longer for the water to freeze, then the explanation would stand that the water's temperature was lowered by the "boiling" event, bringing the water closer to freezing by this temperature reduction. Yet, it seems to follow that the water freezes by the reduction of the gas pressure within as energized particles of matter (the gases) escape out of the water, and the energy of motion of the water molecules is reduced by its exposure to the vacuum, and this allows the formation of ice crystals which yields either a solid state in a beaker or dish (bound by gravity) or a snow-like state suspended in space. Also, as noted in the astronaut's observations, snow-like crystals may form due to nucleation points within the liquid on account of the various proteins found in the urine.
A: The phase shift (sublimation) of water from Ice to vapor occurs in a relatively narrow window. If the ice sample could remain at a temp above a certain level it would all boil away in the vacuum of space.The evaporation and the sub zero temps of space actually cool the sample below the phase shift window thus remain as Ice.
At just below freezing, with a constant pressure (Around 500 mTorr) the evaporation/sublimation process will actually cool the sample out of the window and stop the sublimation process as the system seeks to stabilize. In this system left to its own around 30% of the sample would be turned to vapor and the rest retained as ice when the system stabilized.To keep the process going the Temp or Pressure must be raised. In space BOTH go low and stop the process very quickly.
