The bow is an ancient weapon and people of all social standings - at least the country-dwellers used it often and were probably adept at it.
This is not correct. Literally themain reason the crossbow replaced the bow was that to use a bow effectively required experience and skill, and there were simply not enough people with that skill to fill out armies. Yes, there were forest villages which relied on hunting, and steppe cultures made use bows, but for settled medieval cultures, experienced archers were not widely avaliable.
He pointed out correctly the game they were allowed to hunt: rabbits, fowl, foxes to protect their chicken, squirrels and what not. Noble game like deer, boars and stuff are restricted to nobles by law, right. So unless they are un-free, people usually used bows to supplement their dishes, got food in winter and similar stuff.
Outside of opportunistic kills, peasants did not widely use bows. Again, because using a bow required substantial skill and practice. And because a good bow was a farily expensive piece of equipment. The skills of a bow hunter were completely different from those required from a farmer. Hunting crossbows were popular though.
I remember from somewhere that there even were laws upon what fur commoners were allowed to wear. Do you really think those furs grew on trees (well, squirrel: yes, they do) or that there were specialized huntsmen everywhere who shot animals for fur for the masses?
1. The bulk of common fur could be farmed (i.e. rabbit for both meat and fur)
2. Expensive fur was of course procured by trained huntsmen and traded accross the continent. See for example about the historical sable and beaver fur trade.
About 95% were peasants and worked the fields but it ignorant to believe that plowing and sowing was all those people were able to do. They did carpenter's work when building structures and yes, they went on the hunt if they had spare time to shoot some meat for the cauldron over their fires.
They of course did things other than farming, but was not that common. And most likely hunting was done with hunting crossbows, precisely because bows required much higher skill.
Given that those people usually worked from dawn to dusk, did much work on foot and with manual tools and I believe while uneducated, they at least were generally stronger than nowadays city-dwellers.
Due to the the generally poor diets and poor healt and life expectancy, the average medieval peasant was shorter and not stronger than modern day humans. Doing manual farm work doesn't help muscle growth if you don't have a good diet. And it does nothing good for your back and joints.
Lastly, I cannot believe, that nobles would disencourage their peasants from using bows. If they were free men, they were oblieged to serve in the army in their Lord's wars
This is a misconception. Nobles did not recruit peasants to war (outside of sheer desperation), because untrained peasants were terrible in fighting. A knight was typically expected to bring a small force of semi-professional men-at-arms. Other variations exist, such as fyrd, or byzantine professional troops, but the key point was that random peasant were not conscripted into war.