Unfortunately I can't do every area the justice it deserves, and each really should have its own thread. Also this two-hundred year period is an era of huge changes, covering the height of the Muromachi period to the end of the Sengoku, so one mode doesn't apply to the entire period (or location). I'll do what I can.

1. Each village likely had a blacksmith and carpenter for making the tools and buildings necessary for normal life. In the beginning of the era, outside the cities, urbanization and population wasn't usually enough to support permanent stores and shops. Instead, fairs would be opened a set number of times a month, usually at a local temple, road/crossroad, harbour, or local government office. Merchants would then travel around different fairs. The remnants of these fairs survive in the place names around Japan. For instance, the city of Yokkaichi, means "Fourth Day Fair", because fairs used to be held there on the fourth, fourteenth, and twenty-fourth of each month. In the Kamakura period, three times a month was common. As economy developed in the Muromachi and Sengoku, the frequency of fairs often increased, first to six times a month, then to nine. Note however, the frequency was based on local conditions. Yokkaichi for instance opened as a three-per-month fair in the late Sengoku. With the development of administrative castles and the concentration of samurai population in the castle towns in the mid and late Sengoku, permanent towns with permanent shops became more common, and fairs became rarer. What shops were available probably differed from town to town, but Faris noted the existence of metal workers, carpenters, leather makers, tatami-mat makers, stonemasons, shipwrights, miners, lumbermen, dyers, lacquer painters, roofers, paper workers, silversmiths, potters, plasterers, tile makers, bronze casters, arrow and armour maker. As well, the cotton industry took off in this period, first introduced by import in 1429, by the late 1400s there were cotton cultivated in the Kinai, and 1580s was a common crop in the Kantō. Cotton was shipped to the Kinai to be woven into cloth. I assume it was then carried and sold by merchants.

2. The meaning of the term daimyō changed throughout Japanese history. For more detail on that, please see here by /u/Morricane and I. For most of the time period, a village likely did have a samurai who oversaw it. However, whether the samurai was appointed, and if so by whom, can only be judged case-by-case. In the beginning of this period, they could've been assigned by the shugo (provincial lord), or more realistically the shugodai (deputy-lord). But they could also have got or inherited assignments by aristocrats and temples who were offsite owners. These were people or organizations who had ownership of the land plots (or at least, their taxes) but lived far, far away and so had others collect the tax and send it over. Smaller samurai with a working relationship directly with aristocrats and temples were more common in the Kamakura, and in the Muromachi aristocrats and temples more commonly worked with the shugo, who probably handed the assignment downwards. And of course the shugo had his own land to assign. These mostly worked as tax-farms: a set amount was sent upward, and the collector kept the rest. The village could, of course, be a samurai's direct fief. Meaning he was entitled to the entire tax from there. Fief could've been assigned, but a lot were ancestral, passing down the generation. Often the lord simply confirmed the ownership, many issuing such confirmation every time there's a new lord. With the outbreak of war the previously widespread working-agreement with distant offsite owners really fell away, as local samurai just stopped sending the tax to their owners, and basically became the land's new owners. At the same time, due to war, many local strongmen appeared who straddled the boundary of warrior and not. And the lords didn't much care, since they needed the manpower. So a lot of people oversaw a village just because they did, not because they were assigned. Of course, due to the changing fortunes of war, ownership also frequently changed. For the most part, in the Sengoku there were little material tax movement between government (samurai) levels. Instead a samurai was expected to contribute men and material for assignments, military or otherwise, ordered by the lord roughly proportional to the amount of land production owned. Towards the end of the period though, with the development of castle towns and the movement of samurai there for control and mobilization purposes, more and more samurai were directly paid a stipend from the lord's treasury. In such cases, village elders, not samurai, oversaw villages directly. Such trends continued to increase into the Edo period, but even by the end of the Edo, a not insignificant number of samurai still had their own fief.

3. Prior and during the Muromachi periods, maps were of shōen, since the basic administrative unit was the shōen, and most had multiple villages. There's an understandable lack of such local-level maps in the Sengoku, but some examples of a type of administrative map that roughly lay out equally divided plots of farm land of a shōen, for ownership and administrative purposes especially water rights, does survive. These are called sashizu. However due to their nature you'll be hard-pressed to find more than the main roads, river and ponds, and abstract farm plots from them. See some examples below:

Muromachi shōen ezu (maps): one, two
Sengoku-era sashizu: 1515, undated, undated
Catalogue of some surviving Edo-era village maps here.

here you can also see a slightly earlier period (14th century) village with its ezu and sashizu (actually just a simplified ezu) side-by-side.

5. For most of the period this is actually false. Tax on farm harvest, called nengu, was paid in coinage. Only from the late Sengoku onwards, due to a variety of reasons, did farm tax come to be paid in grain, usually in rice. While the nengu was probably the main tax when averaged across the country, there were plenty of other taxes, tolls, and levies used to get manpower and resources out of farmers and other people. A fishing village might have gave the lord certain catches, help ferry people or transport goods or other duties, made ships, and/or paid taxes in coins. You can read here for more details.