The temperature can go down without there being snow but it cannot snow without the temperature going down.

I think this claim is even more dubious: "Another dataset relates to the cost of rooms and apartments for students and consists of the size of the apartment and the monthly rent. Again it is obvious that the size causes the rent and not vice versa"

I think the mechanisms which set rental prices in a market, and the availability of properties of different sizes to students at different prices, are probably pretty complex functions of supply and demand factors that, themselves, are dependent on behaviors in other markets, like the mortgage market, the higher education loan market, the job market, and so on. You don't even need to talk to an economist; a realtor will tell you the most important factor driving housing costs: location, location and location. That's why two apartments of the same size in New York and Nebraska will have different rental rates.

You can think about causality in terms of intervention. If you make an intervention that changes only the price of an apartment (e.g. you subsidize it), will the size of the apartment change because of that? Obviously not. If you make an intervention that changes only the size of the apartment (and leaves all other mechanisms as they are), will the price of the apartment change? Most likely it will.

If you make an intervention that means a person is willing to pay a little more for an apartment, will they end up renting a larger apartment?

Is price set by the supplier or the buyer? I think an economist would tell you 'both'.

Nice puzzle. Keep in mind that OP is not interested in finding all causes (like the availability of concrete or the size of the rooms), only in checking the causal link between two observed variables. I think the solution is this: to check if there is a causal link from price to size, you need to make an intervention on the price, not on the buyer. (Unless you are sure your manipulation of the buyer can affect the size only through its effect on the price.) The (price, size) dataset in the paper is from a website for students who look for an apartment. Your manipulated student who is suddenly willing to pay more will probably rent a larger apartment, maybe even cause a new large apartment to be built and prices may shift a tiny bit. But you largely failed to manipulate of the price of a particular data point. The apartment that your student would have rented is still in the data set and its price is still largely influenced by the number of rooms, and the chance that it has gotten a room added is so tiny that it won't register in the data. (The causal link is there, I agree, but it's as weak as a butterfly.)

" The fallen snow doesn't just disappear, wouldn't it affect later measured temperatures?"

Unless your weather station is totally covered by snow, no.

And the effect of snow in the total thermal capacity of a city is small. Just compare its mass to the mass of "all things" (buildings, etc)

Snow and ice does have an overall effect on climate, due to its high albedo reflecting light that would otherwise be absorbed and turned into heat.

Why would you be sampling at a single arbitrary point like that?

I'm interested in how things diverge at all scales.

Imagine two identical snowing days, except on one of the days you remove the 6 feet of snow on the mountain. There wouldn't be any difference in the amount of snow that falls? Or the local and global temperatures?

That strange assertion leapt out of the page for me as well. I suppose it depends at what altitude you are measuring the air temperature.

I think it depends on the variance of the temperature across the entire space. If a place has been very cold (below freezing temperature of the water content) for a long period of time, then you might expect all of the water to have precipitated.

However, in many real situations on earth, the temperatures often fluctuate right around the freezing point of the water in the space.