Is this way of writing grammatically correct?

Yes, it’s perfectly correct

It is correct. It is a short practical phrasing that makes sense for an episode summary. 

It’s correct.

Yes

\*something missing

It’s something called an absolute construction. It’s not terribly common in English.

It’s an important construct in the sentence: Buffalo buffalo, Buffalo buffalo buffalo, buffalo Buffalo buffalo.

This is an appositive phrase. Your way is also a valid way of writing it but isn't necessary.

This is description language, like on the back of books, Netflix descriptions, etc. It's similar to headline language, like from news headlines. I don't know if there are full rules, but you should get a feel for them if you just read a lot of them.

Yes. Absolutely correct.

so the way i think of sentences like this is just that since it’s in commas it is unneeded information, i.e. u can remove it and the sentence still makes sense. without it , the sentence is “light orders mikasa to begin killing” but with that extra sentence in the commas just adds more context.

Yes, it's correct. It's a nonrestrictive clause. Surprisingly, prepositions don't have to introduce nonrestrictive clauses as much as many believe.

Commas can be used to ‘pause’ the writing in the form of an interjection. This is called a parenthesis, where information is added to an otherwise grammatically correct sentence in order to add relevant detail. When this happens you can think of the commas like round brackets (). To make things confusing these round brackets are also called parentheses because they are commonly used to create a parenthesis in writing. In creative writing commas are used more often than round brackets because they flow better in writing. To apply this to your example: “Light, his memory recovered, orders Misa to begin killing after Higuchi dies.” You can tell this is a parenthesis because the words in the commas can be omitted: “Light orders Misa to begin killing after Higuchi dies.” The sentence can also be written as: “Light (his memory recovered) orders Misa to begin killing after Higuchi dies.” However in creating writing many authors will avoid this because it looks blocky.

Yep, perfectly normal

I don't love it but it's grammatically correct. The phrase "his memory recovered" stands in for "\[now that\] his memory \[is\] recovered". A more formal way of saying this (and the way I would phrase it if I were writing this) would be Light, having recovered his memory, orders... I like it better because it keeps Light as the subject for both phrases.

It’s a little terse, but there’s no grammar problem here. This is a noun phrase set in apposition to the preceding noun. It amplifies our information about Light, but not in an essential way. You have some leeway with these, for example: Kansas City, “The Paris of the Plains,” was once disparaged as a hive of depravity.

Yup! What they are really saying is: "Now that Light's memory is recovered, he orders Misa....etc"

I’m assuming “Light” is a person? If so, yes, this sentence is fine. Just a bit more flowery than everyday speech.

Yes, it is..still try to adopt technology

It's called "[nominative absolute](https://en.m.wikipedia.org/wiki/Nominative_absolute)".

In Latin it's called an ablative absolute, I don't know what it's called in English but it works. (In Latin it's of course in the ablative tense: "Lux, memoriis reciperatis, Misa iubet ut...")