For the first part of your question, I will refer you here where it has been answered: https://www.reddit.com/r/AskHistorians/comments/1eak5x/what_is_the_difference_between_the_nobility_the/ . The gist of it is that the Gentry were nobles but they were the lowest ranks of the nobility and were not regarded as peers by higher ranking nobles. Essentially they were wealthy individuals but would not have exuded political control that the aristocracy would possess.

A yeoman is a landholder or a freeholder as used in the period you are referencing. They would be considered a commoner, but owned their own property. They did not have the political sway that a noble landowner would possess. Earlier uses of yeoman denoted a household servant in a noble estate. This could range from anything like an usher or a bodyguard. In modern usage the term yeoman denotes varied capability or a sort of "workhorse" skill set. So a "yeoman" tank would be a descriptor for a tank that was capable of solving a wide variety of problems. Cheers!

Post Black Death there was an increasing threat to the hegemony of the aristocracy by the rise of upper-middle and middle class people. Within this period there are three crucial distinctions made that are important to those with social anxieties in maintaining hierarchical order. Firstly, there is the difference of freeholders. Freeholders own a significant piece of their own land, and are the voting electorate of the House of Commons. This is the broadest definition of high society. Then there is the distinction of gentleman. Gentleman need to have a reasonable lineage of wealth, not just posses it themselves. Yeomen would not usually be considered gentleman even when individual yeomen were wealthier than some gentry. Gentry are always gentleman, which is the meaning of the name, they are gentlemen but not aristocrats. Increasingly they are a sort of pseudo-aristocracy which causes resent from the blue-bloods. They draw a third distinction of aristocracy proper, I don't think the answer /uFoamSquared linked is correct in saying aristocrats without titles are gentry, knightly families like the Boleyns were gentry but the second-sons of titular aristocrats were still considered of a higher birth than the sons of merchants even if they had the same wealth. This was a distinction the aristocracy could be zealous about given that all other marks of status could be achieved by gentry the possession of an innately higher social ranks was only more important to maintain.

In one notable example, there was a political rivalry between Robert Cecil and Robert Devereux, 2nd Earl of Essex and to observers this represented the social rivalry between gentry and aristocracy. Robert Cecil's father was entitled, but only relatively recent, the family was of minor Welsh gentry background and Robert Cecil was the second-son so he had no hope of eventually succeeding to his father's new title. In contrast, among Essex's subsidiary titles was 11th Barron Ferrers, so he had a long lineage of aristocracy. And he was considered an aristocrat from the moment he was born. His followers referred to the Cecils' bureaucratic allies, largely from relatively minor backgrounds like themselves, as "goose-quilled gents" (although the Cecils has some notable aristocratic friends). You can see that this distinction occurs throughout the families and not just the entitled male line, because Robert's wife was the daughter of a 10th Baron rather than a 1st Baron and it's notable than Elizabeth I granted her the right to the funeral of a baroness (probably in part because she had such a strong relationship with Burghley Robert's father but it's noteworthy that she was actually considered blue-blooded enough to deserve this). Furthermore Robert's nieces by his sister Anne were significantly more prominent at court than his daughter Frances, even though their own father Edward DeVere was disgraced.