Translation:Each author writes in his own way.
So this is irrelevant to the Abstract Objects lesson, but I've been curious. Is the use of "Autor" and "seine" here considered exclusionary language? If you're talking about a mixed group of male and female authors, would you use this sentence, or something else? And how about the sentence, "Jede Person schreibt auf ihre eigene Weise?" Is that fine to say when talking about a group of men? Am I just bringing my English-speaking hangups into this? Does a German speaker care about these things?
Traditionally, forms like "jeder Autor" or "alle Autoren" can be understood to refer either to male authors only, or, in a more general way, to a both male and female authors. In the latter case, these forms are used as a so-called "generic masculine". Personally, I have no problem with that (I am a woman, BTW).
In recent decades, feminists have criticized what they perceive as sexist language. This criticism is still highly controversial in Germany. However, in some fields such as job adverts the generic masculine is now avoided. Creating a gender-neutral language is more complicated in German than in English, though. For instance, just turning "jeder Autor" into the plural ("alle Autoren") is not enough, as "Autoren" is still grammatically masculine. So you would either have to mention both the male and the female forms (e.g. alle Autorinnen und Autoren; alle Autor/innen; alle AutorInnen, alle Autor_innen) or use a related term that is identical for both genders (e.g. alle Schreibenden). Due to these difficulties, some people - myself included - think that gender-neutral language often leads to an artificial and overly complicated writing style.
As for "jede Person" - as far as I know, nobody has complained about this term so far, probably because it is grammatically feminine. And yes, you could use it to refer to men, although it does perhaps sound a bit stilted if you use it instead of "Mann" when you know you're talking about men only.
I gather the reason that this is so important in job adverts is that the use of male job titles has been demonstrated to dissuade women from applying. This probably works subconsciously, even if the potential applicants would themselves have used generic masculine descriptions.
Weg is more a place or direction and weise more abstract like the way I behave or the way I dance.
I'm not sure of the grammatical accuracy of the singular 'they', but everyone uses it regardless, so I think it should be accepted
No, that's not idiomatic.
PS: Note that both "Weise" and "Weg" (way) have to be capitalized because they are nouns. An uncapitalized "weg" means "away" or "gone".