Okay, the way I understand, the circle of fire beats wind beats lighting beats earth beats water beats fire is that, generally speaking, a when two attacks of equal or near equal strength are used against eachother, the jutsu with a disadvantage loses, meaning it's either cancelled, or reflected (depending on the specifics). But a sufficiently stronger jutsu can still overpower an attack that it should have a disadvantage against, one example being while Sasuke is chasing Temari and Gaara, he uses a fireball, but Temari defends against it with her wind style, because her wind was strong enough to overcome the disadvantage.
The combinations aren't all completely official, which is why some of them don't make as much sense as the others, and the importance of their components is essentially to determine which styles should have an advantage or disadvantage against them. Dust doesn't have much of an explanation for its combination, but as another anon said, it isn't literal dust powers, it reduced things to "dust" and is called particle style in at least one version of the english translations, it's easier to just read the wiki page about it. Yin and Yang are semi-retcons, an explanation from early in part I of what chakra is involves saying that it's a mix of physical and spiritual energy, yang is physical and yin in spiritual, not all, but some techniques that don't have any of the five standard elements are based on those, and all genjutsu techniques are yin by default. Also in the novels there's a mud style, which is earth and water too, meaning the combinations for kekkei genkai styles might not be as limited as the chart implies, so maybe some of them, like magnet and/or explosion might be a repeat combination of another two elements instead.
Any other questions you think I might be able to answer?