Well, the great thing about this topic is that everyone and their Mum has written about it. The bad thing is that everyone and their Mum has written about it, so you get a lot of people arguing.

My sources for this will be 'The Medieval Archer', by Jim Bradbury; 'The Soldier in Later Medieval England', by Adrian Bell, Anne Curry, Andy King, and David Simpkin; 'Knights and Warhorses', by Andrew Ayton; 'Infantry Warfare in the Early Fourteenth Century', by Kelly DeVries, and the reply to some of DeVries's claims on the efficacy of the longbow by Clifford J. Rogers.

Summarised: an archer was typically a semi-professional soldier with a secondary trade he could fall back on, though there were some who were entirely dedicated to warfare as a career depending on the period. Some archers became men-at-arms, others served as archers during the entirety of their military careers. The majority of them came from the peasantry, with a larger proportion coming from the upper strata of this amorphous 'class' during the Hundred Years War.

Answer to 1: It is entirely possible to work a 10 hour day and develop another skill to a high level of proficiency. I fear you're making assumptions about the majority of medieval people that I think a lot of people have. In general, even the lowliest peasant had atleast one day free of work, Sunday. They did not live entirely hand to mouth, and many had time for recreation and socialising even on a busy day. Many of the Statutes requiring archery practise stipulate it occurring on a Sunday. The intricacies of acquiring this skill are a little unsure, but due to the extreme musculature that the evidence suggests, I'll mythologise it a bit.

• You take your 6 year old son out to the butts one day and you give him a little bow with 20 little arrows and you teach him to shoot.
• You keep him shooting 200 arrows a day, every day, and every year or so, you give him a bigger bow.
• By the age of 18, you have a grown man who can comfortably sit into a 160 pound war bow and shoot arrow after arrow for hour after hour, with barely a miss.

It is a skill that requires a great deal of dedication to master, but that dedication was well within the grasp of the majority of the population.

Answer to 2: A longbowman, from what we can discern, was typically part of a 'company' or 'retinue' for the majority of the period. We don't know how they were recruited specifically (we don't know if they went into pubs and asked who could hit a target at 200 yards) but the actual requirements and arrangements we have some context for. The king would agree to contract so many men from a lord or noble, and then that nobleman would go out and recruit the number agreed. These would form part of his retinue, and he would serve for a specific amount of time for a specific wage with specific equipment. This is called the 'indenture' system, and it is as complicated as my ex-girlfriend's reasons for leaving me. Men would defer wages, companies would serve as 'sub-contractors' within retinues, the number of archers could be increased if too few men-at-arms could be found. The reasons for archers enlisting are quite simple; it was bloody good business. 6p a day was more than a stone mason earned, and they also had a high chance of acquiring financial gains via looting. The other motivations you mentioned, adventure and patriotism, also played a part. Many archers were pardoned criminals; indeed, perhaps a third of Edward III's 1346 army consisted of criminals contracted to serve as payment for freedom. We cannot find much evidence that archers of this period were conscripted to serve in France. There were not, not, NOT feudal levies. The commissions of array associated with feudal levies died out during the 14th century, with the last use of a feudal call to arms being by Richard II in 1388, and this being the first time since 1327 that it had been used.

Answer to 3: Some were maintained in permanent retinues of nobility, others became mercenaries in other countries, like you say. Most went home and became foresters, farmers, hunters, stewards, bailiffs, guards, and a hundred other professions.

One thing I want to note is that the period you've described encompasses a large degree of social, political, and military change in England. Most of the things I say are true by the early 15th century, while others are untrue by the late 15th. Many men serving in Normandy between 1421 and 1453 can very well be considered 'semi-professional' soldiers, though we run into difficulties with the definition of 'professional'. Some archers earned a living fighting in the campaigns of 1337 to 1345, while others only ever partook in the 1453 Castillon campaign and never returned. With the Black Death changing the fabric of how the peasantry interacted with the nobility, England of Henry VII had completely changed in a thousand ways from that of Edward I. The archers who fought at Falkirk were almost entirely levied by commissions of array; they were paid for their time, but they were conscripted all the same. Those who shot alongside Henry V at Agincourt were highly experienced veterans who served overseas for pay, employing themselves as soldiers.

I hope this answers your questions!

Edit: formatting.

I have written a couple of previous answers about the demographics and recruitment of English archers here and here that you may find useful.

The short of it is, most popular notions of the "typical" medieval army or "typical" medieval warfare are extremely inaccurate. While medieval Europe did have a highly stratified society, it was not the case that one was either a decadent aristocrat or a serf scrabbling in the muck. There were plenty of people who were of means and owned significant amounts of land but did not have noble titles, or who were skilled urban craftsmen or small landholders who owned their own fields. These are some of the prime demographics for medieval military recruitment: the middling ranks of society who could afford to equip themselves for war (and practice martial skills), but still hungry enough for wealth, power, and station that the life of an ordinary soldier would tempt them. English archers in particular were often second sons or younger cousins from military families, who would equip their oldest son with heavy gear that made him eligible to serve on horseback as a man-at-arms. Often these families lacked the means to afford multiple sets of this expensive equipment, and so the younger men of the family would fight as archers. They would have had plenty of time to practice, not only with the bow, but also with the sword and buckler (contrary to popular belief, in the period we are discussing, a sword was a relatively affordable item).

Conscription into service did happen, either as part of the local array of a shire or as a criminal serving their sentence through combat, but plenty of soldiers were volunteers. But plenty of soldiers served as volunteers. For one, the pay was quite good for the era, even as a common archers, even without taking into account the question of looting and ransom. Additionally, as I mentioned above, many of these soldiers came from military families and were part of a broader English military community. Going to war was, for many, simply what they did. Unfortunately, period sources are not very engaged with the inner thoughts and motivations of ordinary soldiers, for the most part. So while we can discuss many of the factors that might have shaped the lives of these archers, we don't have memoirs or similar recollections from the soldiers themselves.

Archers were perhaps more often professionals than you would think, especially after English conquests in France created a pressing need for more permanent garrisons. However, many of them were more accurately described as "semi-professionalized." They would fight in multiple campaigns throughout their lives, but when there was no immediate war, they had their regular occupations. As described above, many came from the skilled trades or families that owned land. Many archers serving in aristocratic retinues were not full-time soldiers, but craftsmen employed by the king or a duke. A "John Tailor" might spend his days tailoring clothes, but then serve as an archer when the king went to war. The "household in arms" was not just an expression. It was literally the king's household taking up arms when the king went to war. Some archers simply became professional mercenaries and fought in other parts of the world when England had no active employment for them. Sir John Hawkwood is probably the most famous English mercenary of the 14th century and earned enormous wealth through his rampages across Italy.