It's a bit of a tricky question as just defining "average person" and "Ancient times" is difficult as it would depend greatly on which civilization and during which period. But let's say somewhere within the last few centuries BCE, and let's assumed this person has a basic education or lifestyle.

In that case: yes. The changes of climates were well known to most educated individuals, Plato wrote about it, and Aristotle wrote an entire treatise on it called Μετεωρολογικά (Meteorologica), and we know people in Persia understood the physics of water, and even had ice. https://en.wikipedia.org/wiki/Yakhchal

Trade and warfare brought people in touch with cultures far outside their own as well. So many soldiers would have experienced different climates and when you consider elevation changes as well the idea of snow and desert are not always separated by great distances. The Kurdish mountains for example get snowfall.

One of the problems in answering this is that "the average person" in ancient times in the desert, at least in the region I study, was illiterate and left no record of his meteorological views. That being said, while unusual, snow actually occurs from time to time in a perhaps surprising range of Middle Eastern desert areas. There was snow a couple years ago in Cairo and Northern Saudi Arabia. The mountains of the United Arab Emirates get snow from time to time. Moreover, a very large proportion of the population of the Arabian desert were nomadic or semi-nomadic, often between seasonally migrating between highlands (that might get snow) and lowlands. Others engaged in trade across rather extensive distances. So a copper miner in ancient Oman would have been linked by trade with pretty much the entirety of the Persian Gulf. The Hejazi coast was closely linked with the Levant, which gets quite a lot of snow.

So while we can't say with any certainty given the absence of sources, I would be quite surprised if the answer was not "yes" to your first question.