A lány is jegyzetel, hogy otthon is jól tanuljon a vizsgára.

No. Hungarian is means also / too / as well.

In this sentence it appears twice:

• A lány is = the girl too / the girl also
• otthon is = at home too / also at home

A very important point: Hungarian is comes after the word or phrase it belongs to.

So:

• A lány is jegyzetel = The girl also takes notes
• otthon is = at home too

It has nothing to do with the English verb is.

Jegyzetel means to take notes or to make notes.

It is related to jegyzet, which means note or notes, but jegyzetel is a full verb on its own.

Here jegyzetel is:

• present tense
• 3rd person singular
• roughly she takes notes / is taking notes

So the first part means that the girl is taking notes.

Because here the verb is being used without a definite direct object.

Hungarian often distinguishes between:

• indefinite conjugation
• definite conjugation

In this sentence, the girl is just taking notes in general. There is no specific object like the notes that she is acting on. So Hungarian uses the indefinite form:

• jegyzetel

A form like jegyzeteli would suggest something more like she is noting down it / the specific thing, which is not what this sentence is doing.

Hogy introduces a subordinate clause. In this sentence it means something like:

• so that
• in order that

So:

• jegyzetel, hogy... = she takes notes so that...

Hungarian uses hogy very often in these purpose clauses, even when English might use a simpler structure like to study well at home.

Because after hogy in a purpose clause, Hungarian often uses the subjunctive / imperative form.

So:

• tanul = she studies
• tanuljon = that she study / so that she may study

In natural English, this is usually translated more smoothly as:

• so that she can study well at home
• in order to study well at home

So tanuljon does not sound like a command here. It is expressing purpose or intended result.

Because otthon is already an adverb meaning at home.

Hungarian has some location words that work as complete adverbs by themselves, and otthon is one of them.

So:

• otthon = at home

You do not normally need an article or an extra ending here.

If you tried to build something more literal like in the home, that would sound different and much less natural in this everyday meaning.

Vizsga means exam.

A vizsgára means for the exam.

The ending -ra / -re often has a directional meaning such as onto / toward, but with some verbs it is used in an idiomatic way. With tanul, Hungarian commonly says:

• tanul valamire = study for something / prepare for something

So here:

• a vizsgára = for the exam

The article a shows that this is a specific exam.

Because jól is the adverb, while jó is the adjective.

• jó = good
• jól = well

Here it modifies the verb tanuljon, so Hungarian needs the adverb:

• jól tanuljon = study well

This is the same basic difference as English good vs well.

Yes, Hungarian word order is flexible, but it is not random. It often shows topic, emphasis, and what the speaker wants to highlight.

In:

• A lány is jegyzetel...

the phrase A lány is sets up the girl too as the topic/additive element.

In the second clause:

• otthon is jól tanuljon

otthon is means at home too, and jól stands right before the verb, which is a very common position for an important adverb.

So the order helps show what is being highlighted:

• the girl also
• at home also
• study well

Also remember: is must come after the word or phrase it belongs to, so its position is especially important.

Because Hungarian often leaves out subject pronouns when they are already clear from the verb form and the context.

Here tanuljon is a 3rd person singular form, and it is obvious that the subject is still the same girl.

So Hungarian does not need to say ő unless there is some special emphasis or contrast.

This is very normal in Hungarian.