The river is freezing very quickly, however moments later it's raining instead of snowing.
If the temperature at the crash site was indeed -60º Fahrenheit, the Robinsons would have suffered frostbite in 7 to 10 minutes, without any additional factors like wind chill coming into play. Even in the unlikely event that they did not die of hypothermia after 5 hours in that temperature, they would likely have lost their ears, noses, fingers, and large swathes of their exposed skin to permanent tissue damage.
Magnesium is far too reactive to appear in pure form in nature. In it's ore forms it would not burn as shown or be much use to the Robinsons.
As the ship is crash landing, the ship is calling out the altitude, and a shot of the display on the ship is shown. The voice is calling out distance in feet, the display is showing distance in meters, however they're using the same number, for example, when the voice says 3000 Feet, the display shows 3000M (for meters), which would be over 9,000 feet.
When water freezes it releases about 334 kilojoules (or about 93 watt-hours) of energy per liter. If we assume the body of water that froze to be about 1 thousand liters (a 10 meter by 10 meter by 10 meter volume) that would be 93 megawatt hours of energy or the average daily energy use of about 3,000 homes. This energy was dissipated over about a minute, which means the pond would radiate about 93 megawatts of power every second. large amounts of water in the real world, even at very low temperatures, freezes slowly. The reverse is also true: ice melts slowly unless exposed to high temperatures. This is why ice in a drink keeps the drink cool until the ice is almost entirely melted, after which it relatively quickly reaches ambient temperature.
Within the first couple of minutes of the episode as the Robinsons play cards in zero gravity, Maureen Robinson takes a card from the deck with one hand. The deck is held together with an elastic band and does not move, defying Newton's Laws of Motion.
In the first scene, while the family plays "Go Fish" at the table, the deck of cards is floating freely in zero-G, but when the alarm goes off and they are told to put their helmets on, all members show some level of strain picking them up. Although the helmets have mass and might be difficult to move at first, they should be easy to maneuver in zero-G once they are in motion.
In the opening scene, as the Jupiter 2 crashes out of orbit, the instruments display the altitude in meters. The computer voice incorrectly announces the same figures in feet.
In the opening crash sequence of "Impact", the computer voice states loss of altitude in feet, but the computer's display is showing meters.
Everything about the temperature around the lake makes no sense. The speed the water freezes when Judy is in the ship suggests a source of EXTREME cold beneath the floor of the lake. Nothing natural could do this underground without breaking the first law of thermodynamics. The lake should freeze from the top down and at a much slower rate. Also, whatever caused such a rapid freeze would also drop the air temperature above it to where you could not have exposed skin for more than a few seconds without issues and minutes without frostbite. Rain would be completely impossible and even snow highly unlikely at those temperatures.
The magnesium mountain is far too far away to reach in the time they have, let alone get back. John is meant to be career military and would know this immediately. Even if he were desperate enough to try the journey taking Will along would easily halve his speed over such harsh terrain.
Using magnesium to melt the ice would possibly work (though it would need a lot more Mg than John brings back) but with the unfortunate side effect that it would certainly kill Judy. Her suit would never withstand the heat and once breached the slurry of metal and water would pour in causing a particularly gruesome death.
The ship somehow appears to be fully functional below the surface even when full of water. This would have to be a deliberate design goal and surely something the crew would know about. Leaving the ship would be the last thing they would do.
Although no time period has been set, John can be seen using an iPhone 6 when talking to Maureen about custody.