I was playing the role of Chaucer's yeoman for April Fool's, so I can answer a few of your questions. My characterization of him was based on (besides his description in Canterbury Tales, obviously) a quite excellent article by Kenneth Thompson, "Chaucer's Warrior Bowman: The Roles and Equipment of the Knight's Yeoman." Much of previous Chaucer scholarship that touches on the Yeoman has tried to emphasize one role over another, with someone suggesting that the Yeoman is supposed to be a "man of peace," which is a hilarious contrast to the amount of weapons he's hauling around. In reality, Thompson correctly shows that the Yeoman, much like his real-life counterparts whom Chaucer would have have known, has a variety of roles that he is suited for and would be expected to handle. "Yeoman," at the time Chaucer wrote his Tales, meant "servant" and could be applied to those who occupied a kind of middling place in society.

His "day job," so to speak, is as a forester. His most important task there would be to ensure that the deer in his forests are maintained for the hunts held by the Knight. He would be responsible for not only preventing poachers from killing too many of the deer, but also feeding the deer in winter and take care of newly-born calves. In addition to taking care of the deer, a forester also prevented illegal grazing and logging in the woods. So law enforcement, animal husbandry, and resource management are all things he would be doing in the course of his normal duties. He would also participate in the hunts as an assistant. His role in the hunt would be determined by the kind of hunting being done. Presumably he would be an archer or a guide of some kind; nothing in his description would imply that he works with the hounds.

As I implied in one of the follow-up questions about Chaucer, there was a difference between the senior administrators of the forest (who might be called "forester" in their official job title) and the guys actually in the forest doing the daily work. Chaucer indicates that his Yeoman is a practical guy: he's got a tan because he's outdoors all the time and "of wodecraft wel koude he al the usage." He's equally good with his weapons: "wel koude he dresse his takel yemanly." He knows how to maintain his equipment and he has enough money to buy the good stuff. Aside from his "gay daggere" and "gay bracer," his arrows are fletched with peacock arrows rather than the common goose. The actual utility of peacock-fletched arrows is debatable, but they were often regarded as superior to goose. Chaucer is indicating that the Yeoman can afford the best arrows available. Presumably the Knight is paying him a good salary or he's acquired a lot of plunder on campaign.

There's actually nothing directly in Chaucer that would indicate that the Yeoman was a soldier at some point. Unlike the Knight (a valiant crusader) and the Squire (a mounted man-at-arms who has participated in chevauchées), the Yeoman's combat record is not mentioned. However, we can presume that he knows how to fight. He's well-armed, with bow, sword and buckler, dagger, and peacock arrows. The most likely reason for him to be on the pilgrimage to Canterbury is that he is acting as a bodyguard for the Knight and Squire. Neither of them are mentioned to carry arms, although they almost certainly are both bearing swords and the Knight is wearing armor. The language Chaucer employs to describe the Yeoman and the focus on his equipment function to describe the Yeoman himself as a kind of weapon, wielded by the Knight and the Squire. Again, there's nothing specific to indicate that the Yeoman has ever been a soldier. He has no tale or dialogue that would help clarify his past. But in the context of the other two men he's with, military service is strongly implied. Not all foresters were necessarily soldiers, but there's an obvious overlap in the skills associated with both roles. When medieval aristocracy went to war, their households served and fought alongside him. Looking at the pay records of campaigns like Henry V's 1415 invasion of France reveals names like "John Tailor" and "Richard Armorer." These occupational surnames indicate their roles in the peacetime household, but in war, they were mounted archers.

Together, these three men are a microcosm of the English army in Chaucer's day: the knightly commander, the man-at-arms, and the regular mounted archer. All of these things together indicate a martial role for the Yeoman. His service is not mentioned directly perhaps because Chaucer doesn't feel he needs to say that the Yeoman accompanied either the Knight or the Squire to war. It seems to be the case that the Yeoman is a trusted figure to them: he is the only servant they have with them, no more are necessary. This makes sense with the evidence we have of English military recruitment in the Hundred Years War: men going to war often did so in the company of tight-knit groups of local family or friends, which would then join with larger companies to form the army as a whole. These friends and family members did not necessarily fight all in the same role; a man-at-arms might be accompanied by his brother serving as an archer.

I don't have any idea about foresters outside of England, unfortunately. You'll have to ask someone else about that.

Sources:

1. Kenneth Thompson, "Chaucer's Warrior Bowman: The Roles and Equipment of The Knight's Yeoman," The Chaucer Review, Vol. 40 No. 4.

2. Richard Allmond and AJ Pollard, "The Yeomanry of Robin Hood and Social Terminology of 14th Century England," Past & Present No. 170.

I can speak very briefly to point #3.

Similar systems of forest management are seen in Germany and the Netherlands into the 19th century. They are generally called "jaagers" (hunters) and were professional woodsmen and game wardens who worked in the employ the local nobles who owned the hunting rights to a given piece of woodland. The local jaager of an area might work as soldiers, but seem to have often take up trades like charcoal-burners or sometimes lumberjacks if they could not find work in the employ of the local notables.