Новая грамматическая тема: конструкция «едва ..., как ...» помогает сказать, что одно действие произошло сразу после другого.

It is a two-part sentence pattern:

• едва + clause 1
• как + clause 2

It is used when the second action happens immediately after the first one.

A typical example:

• Едва он вошёл, как зазвонил телефон.

This pattern is similar to English hardly had he entered when the phone rang or as soon as he entered, the phone rang.

Yes. On its own, едва often means barely / hardly:

• Я едва вижу. = I can barely see.

But in the fixed construction едва ..., как ..., it has a more grammatical function and means something like:

• hardly ... when ...
• as soon as ...

So the word keeps its original flavor of hardly, but in this pattern it expresses immediate succession of events.

Because this is a complex sentence with two clauses.

In Russian, when one clause is introduced by a conjunction like едва, it is normally separated by a comma. Then как introduces the second part of the pattern.

So:

• Едва начался дождь, как все побежали домой.

The comma is standard punctuation here.

In the named pattern едва ..., как ..., yes, как is the expected second part.

That said, Russian has related alternatives, for example:

• как только ...
• едва только ..., как ...
• sometimes simply едва ...

But if you are specifically using the textbook construction едва ..., как ..., then как belongs there.

Because after помогать / помочь, Russian often uses an infinitive to show what action is being helped.

So:

• помогает сказать = helps (someone) say
• помогает понять = helps (someone) understand
• помогает запомнить = helps (someone) remember

In this sentence, the subject is конструкция:

• конструкция ... помогает сказать ...

Literally: the construction helps say / helps express ...

Because the subject is одно действие.

The main noun here is действие, and действие is:

• neuter
• singular

So the past tense verb must also be neuter singular:

• действие произошло

Compare:

• случай произошёл — masculine
• встреча произошла — feminine
• действие произошло — neuter

После requires the genitive case.

So:

• после чего? = after what?

That is why we get:

• после другого

More fully, this means:

• после другого действия

The noun действия is omitted because it is already understood from одно действие earlier in the sentence.

So the full underlying idea is:

• одно действие произошло сразу после другого действия

Russian often omits repeated nouns when the meaning is clear.

Because одно ... другого is a very natural Russian way to talk about one thing and the other one in a general sense.

Here it means:

• one action happened immediately after the other

Using первое and второе would sound more like you are explicitly numbering them. That is possible in some contexts, but here одно ... другого sounds smoother and more natural.

It is fully correct and standard, but it can sound a little more literary or written than some simpler alternatives.

In everyday speech, many speakers often prefer:

• как только ...
• сразу после того как ...

So:

• Едва он пришёл, как все замолчали. — correct, a bit more stylized
• Как только он пришёл, все замолчали. — very common in everyday speech

A learner should definitely understand едва ..., как ..., but does not need to use it all the time.

Because Новая грамматическая тема: works like a heading or label that introduces what comes next.

It is similar to English:

• New grammar topic: ...

So the colon signals:

• now the explanation follows

This part is not really the main grammatical example itself; it is more like a classroom or textbook introduction.

Because « » are standard Russian quotation marks in printed writing.

So in Russian typography, you often see:

• «слово»
• «едва ..., как ...»

English usually uses:

• "word"

Russian usually uses:

• «слово»

Here the construction is placed inside quotation marks because the sentence is mentioning the expression itself as a grammar item, not using it in a normal sentence.