OK, I'll take the bait.
TL;DR: the famous netherese arcanist Avogadro Wished anyone to die after spending 10 minutes in a Bag of Holding in -1,013 DR and his spell still continues to take effect to this day.
The volume of Bag of Holding is 64 cubic feet. Feet is a fantasy unit D&D uses to make things sound medieval

and 64 cubic feet are equal tpo 1,812 litres in the real world.
A man breathing for 10 minutes would consume oxygen (21% of air on the intake, 16% on the way out) but it's not lack of oxygen that would cause harm to the denizen of the bag -- the human could rebreathe air a few times before there isn't enough O2, it's the increased concentration of CO2. The minimal amount of O2 (at sea level) is around 10%, and 6-8% is lethal. The first risk encountered is the toxicity of CO2. It's 0.04% of the air inhaled, but 4% of the air expired. Rebreathing air with a 4% concentration of CO2 is enough to increase heart rate, 5% headache, 10% is enough for inconsciousnes.
Now, a human is breathing 450 litres per hour if just standing (much less if sleeping, as low as 250, so ability to sleep quickly -- maybe magically -- should extend the time one can hide in a bag of holding). I assume the creature in the bag stays still to reach the 10 minutes. That's 75 litres, much, much less than the amount present in the bag at normal pressure.
An average human standing still exhales 18 litres of CO2 per hour. Over the course of 10 minutes, that would be 3 litres, increasing the CO2 from 0.04% to 0.2%. Not enough to create any significant toxicity.
So, the Bag of Holding must have a much lower pressure than regular air -- opening interesting possibility to increase food preservation while travelling with rations in a Bag of Holding. Our standing regular human would need to consume 90 litres of O2 per hour, or 9 litres in 10 minutes. That's quite low. Really, the amount of oxygen can't be the problem. It's what is present in rought 45 litres of air, over 1812 litres of the bag, so the bag would be... mostly empty of gas, with a pressure of roughly 25 hPa. Which isn't enough. At this pressure, water boils below the body temperature. You won't start to boil (the internal pressure of the body is enough) but saliva, tears, the humidity inside your lungs... will. It's not pleasant (as in, you die). So, the death at the 10 minutes mark shouldn't come for lack of overall O2, but from the effect of low pressure of O2 (as you rightly supposed).
Airline industry provides us with a chart for TUC (
time of useful consciousness) according to altitude. You don't need to be unconscious to be disabled. In the event of rapid decompression (such as would be experienced by someone entering the low-pressure environment of the Bag) it's roughly 5,500 meters to be disabled after 10 minutes -- less if you exercized before entering hte bag. At this altitude the air pressure is arounds 500 hPA, half the sea level pressure. Such an environment is compatible with the effect of the bag, except that the stay in the bag wouldn't be fun at all, but the RAW don't mention it specifically. Why is this pressure constant, despite air being replenished between stays (if not, CO2 level would rise over several stays to become instantanouesly toxic) is left as an exercise to the reader.
What makes the situation wholly magical is that you can breath 10 minutes in the Bag of Holding... even if it is filled before with 1720 litres of water (the remainder of the volume being taken by you).
Note on the OP's question: I rule that object in the bag of holding are transported with no motion in a gravity-less planar bubble. That's why things inside the bag, like a pointy stick, can't damage the bag itself. So the pot of stew wouldn't spill, it would just stand there, motionless, until recalled. I would tentatively rule that it doesn't stay warm longer than it would if it was outside the bag, but I am not certain of that. I fear players pouring lava into their bag and asking it to empty itself at the foot of enemies if heat transfer wasn't a thing in the bag.
Edit: Naah, scrap it. I read again about the bag. It can't be an explanation involving pressure, since the time they survive is 10 minutes divided by the number of creatures. So 10 creatures would suffocate after 1 minute, which wouldn't be the case if the bag just had a lowered air pressure. There would still be enough air for the people inside to breathe for