Adding salt to water makes it freeze at a lower temperature. This fact is being used in two different ways in the two scenarios you mention. Dissolving sodium chloride in water is slighly endothermic, but this effect is small and to the best of my knowledge isn't important in the drink cooling process.
Putting salt on the highway is quite straightforward: we don't want ice to form, so we put salt in the water to prevent that. This doesn't just change the amount of time it takes ice to form, it actually completely prevents ice from forming, unless the temperature gets so low that the water can freeze even with salt in it.
Cooling your drink is a bit more complicated, because in this case the rate at which things happen is important. You don't want your drink to be less than $0^\circ C$ because it would freeze; instead you want to cool it down to a few degrees Celsius nice and quickly.
The rate at which it cools depends on two things: the temperature of its surroundings (the colder the better) and the heat conductivity between it and them. You could try to cool it by putting it in a bowl of ice at $0^\circ C$, but the problem is that the ice is solid and will only touch the bottle at a few points. This results in a poor thermal conductivity, so the drink will only cool slowly.
To get around this, you could try mixing the ice with some water. Now the bottle is touching the liquid over a large surface area, and the liquid itself has a higher thermal conductivity than solid ice due to mixing, so heat will be transferred much more quickly. But the problem is that the water won't be at zero degrees any more, at least not at first (I'm assuming the water comes from a tap, so it's not chilled initially). You have to wait for quite a bit of the ice to melt before the water's temperature will drop. Also, once you put your warm drink into the water it will heat the water up as the drink cools down, so again you have to wait for the ice to melt in order for the water to cool again.
The solution to this is to make the ice melt faster. You can do this by adding salt. This lowers the freezing point, making the water less "happy" about being in the liquid state, so it melts more quickly. This means firstly that the tap water you've added to the ice will cool to close to $0^\circ C$ much more quickly, and secondly that once you've put your drink in there the water will stay cold as the ice continues to melt.
It's also possible that, with the salt added, the water can go to below $0^\circ C$, but this will only happen if the ice is quite a bit colder than $0^\circ C$. This could be the case, but my intuition is that the rate of cooling due to the ice melting faster is more important here than the final temperature. You could easily test this by putting a thermometer in the salty ice water and seeing if it goes much below freezing.
There's also the fact that dissolving the salt is endothermic, as you mentioned. To test whether this is important, you could try adding salt to some chilled water without any ice, and see if the temperature drops a lot. My feeling is that it will only drop by a tiny amount that will be hard to measure with a normal kitchen thermometer, but you can always try the experiment.
The short answer: Cloudy ice is caused by gases (mainly nitrogen and oxygen) dissolved in the water that come out of solution when the water freezes. The small bubbles trapped in the ice cause the white appearance. Boiling the water removes the air dissolved in it, producing clear ice as a result. Assuming that other impurities don't produce the same cloudy effect.
The long answer:
Impurities present in water:
Gases: Water at 20°C normally contains about 15 ppm dissolved gases, which is the equivalent of 1 volume of air per 50 volumes of water. These are the same gases present in air, but not in the same proportions since some are more soluble than others: it's about 63% nitrogen, 34% oxygen, 1.5% argon and 1.5% carbon dioxide.
minerals: Tap water contains dissolved minerals, mainly Ca and Mg. They can be present in the form of bicarbonates: $Ca^{2+}({HCO_3}^-)_2$ and $Mg^{2+}({HCO_3}^-)_2$ (these only exist in solution, not as solid substances), and as calcium and magnesium sulphate. If the water passed through a water softener, the Ca and Mg ions may have been replaced by (twice as many) sodium or potassium ions.
The effects of heating the water:
removing dissolved gases: higher temperature favors endothermic reactions (Le Chatelier's principle). For the gases present in water, dissolution (at room temperature) is an exothermic process, so their solubility decreases when the water is heated. The solubility of gases doesn't reach zero at boiling point, nor does it necessarily decrease over the whole temperature range. For nitrogen in water, the enthalpy of dissolution becomes positive around 75°, and its solubility increases above that temperature.
At 100°C, solubility of air as a whole is $0.93 * 10^{-5}$, about half the solubility at 10°C, $1.82 * 10^{-5}$.
removing dissolved minerals: Heating the water promotes the conversion of soluble Ca and Mg bicarbonates to insoluble carbonates ($2 {HCO_3}^-$ → $CO_3^{2-} + H_2O + CO_2$) which will come out of solution (as limescale). The sulphates (sometimes referred to as "permanent hardness"), and the sodium or potassium (bi)carbonates stay in solution.
The effect of boiling:
- Solubility of gas in liquid not only depends on temperature, it is directly proportional to the partial pressure of the gas. When boiling, the gas phase in contact with the water is no longer the air, but the water vapor (in the bubbles and close to the surface). In those bubbles the partial pressure of the gases will be close to zero, so gas molecules will still leave the liquid phase (and the increased surface area and the movement of the water speeds up the process), but hardly any will return. Given sufficient time, the water vapor will remove most of the gas. Boiling is basically the equivalent of degassing by purging: removing a gas (oxygen usually) from a solvent by bubbling an inert gas through it.
How do gases make ice "milky/cloudy"?
- During freezing, the ice layer starts at all sides of the cube and grows inward. Water molecules fit the crystal lattice and will adhere to it, other molecules won't (but if the ice grows faster than the gas molecules can diffuse away, they will get trapped). The concentration of gases (and other impurities) in the remaining liquid rises, the solution becomes supersaturated, microbubbles start forming. All these get trapped in the ice, giving it a milky appearance.
Best Answer
By shifting the melting point to a colder temperature, the surface of your salt water/ice will be below $0C$. Assuming the environment is warmer than that, the temperature difference will be greater with the salt water ice bottle than the pure water ice. This implies greater heat transfer and faster melting.
If you want it to melt more slowly, the simple answer would be to put a layer of insulation around the bottle. The rate of melting is related to the rate of heat transfer. This means your desire to melt as slowly as possible is equivalent to wanting it to cool the environment as slowly as possible. Insulation will accomplish this.
You can think of your bottles as having almost a set amount of cooling power, rather like the amount of energy in a battery. Adding salt doesn't increase the amount of cooling power. You can slow down how fast the battery is used, but that doesn't make it more useful. Here you can slow down how fast the ice melts, but that might not make it cool your aquarium more efficiently.