Learn Hardcore Finnish: He auttavat toisiaan läksyissä.

he auttavat = they help (standard, written, people)
ne is also they, but:
- traditionally for things and animals,
- in everyday spoken Finnish, used for people too: ne auttaa.

So:

Standard written Finnish: He auttavat toisiaan läksyissä.
Very common spoken Finnish: Ne auttaa toisiaan läksyissä. (verb often in singular in speech)

If you are doing school-type or formal Finnish, use he with -vat/-vät: he auttavat.

Why is the verb auttavat and not auttaa?

The verb must agree with the subject in person and number.

The infinitive is auttaa (to help). In the present tense:

minä autan – I help
sinä autat – you (sg) help
hän auttaa – he/she helps
me autamme – we help
te autatte – you (pl) help
he auttavat – they help

Because the subject is he (they, plural), you need the 3rd person plural form auttavat, with the ending -vat (or -vät after front vowels).

What exactly does toisiaan mean, and how is it formed?

Toisiaan means each other.

It comes from the word toinen (the other, another).

Steps:

toinen → stem toisi-
Partitive plural of toinen is toisia (others).
Add the 3rd person possessive suffix -an → toisiaan.

So literally it is others-of-them, but its normal meaning is each other.

Examples with other persons:

Autan toveriani – I help my friend.
Autamme toisiamme – we help each other.
Autatte toisianne – you (pl) help each other.
He auttavat toisiaan – they help each other.

What case is toisiaan, and why is that case used here?

Toisiaan is in the partitive plural and has a 3rd person possessive suffix.

Structure:

base: toinen
partitive plural: toisia
- possessive suffix: toisia
  - an → toisiaan

The important point: auttaa normally takes its object in the partitive case:

auttaa lasta – to help a child
auttaa ystävää – to help a friend
auttaa toisiaan – to help each other

So toisiaan is the object of auttavat, and because auttaa wants a partitive object, toisiaan must be in the partitive.

What is the difference between toisiaan and toistensa?

Both relate to reciprocity, but they are different forms and used in different structures.

toisiaan
- partitive plural object
- translates as each other
- used directly as the object of a verb
- example: He auttavat toisiaan. – They help each other.
toistensa
- genitive plural with possessive suffix
- translates as each other’s
- used as a possessive determiner before a noun
- example: He korjaavat toistensa virheitä. – They correct each other’s mistakes.

So:

They help each other → He auttavat toisiaan.
They correct each other’s mistakes → He korjaavat toistensa virheitä.

Where is the word their in this sentence? Why is it just läksyissä, not something like heidän läksyissään?

Finnish often leaves possession implicit when it is obvious from context.

Läksyissä by itself can naturally be understood as in their homework here, because the sentence is about he (they) helping each other. There is no need to repeat whose homework it is.

You could make the possession very explicit:

He auttavat toisiaan heidän läksyissään.

This is grammatically fine but sounds more heavy and is usually unnecessary unless you need to contrast:

He auttavat toisiaan heidän läksyissään, eivät opettajan tehtävissä.
They help each other with their homework, not with the teacher’s tasks.

In ordinary sentences, läksyissä alone is enough to convey their homework.

Why is läksyissä in that form? What case and number is it, and what does it add to the meaning?

Läksyissä is:

plural
inessive case (the -ssa/-ssä case meaning in, inside, in the context of).

Base word: läksy (a homework task, a piece of homework)
Inessive plural: läksyissä – in (the) homework tasks.

Here läksyissä expresses the context or domain of the helping:

He auttavat toisiaan läksyissä.
Literally: They help each other in the homework, i.e. with their homework or when they are doing their homework.

Inessive is often used like this for activities or situations:

tunnilla – in class
töissä – at work
kokeessa – in an exam
läksyissä – in/with (their) homework

Why is läksyissä plural? Would läksyssä (singular) or läksyjä also work?

Using the plural läksyissä is the most natural, because homework usually consists of several tasks. Finnish often uses plural for this type of activity.

läksyissä – in the homework tasks (plural, inessive)
läksyssä – in the homework task (singular, inessive) – would sound like there is just one specific task
läksyjä – some homework (partitive plural)

You could say:

He auttavat toisiaan läksyjen kanssa. – They help each other with the homework.
He auttavat toisiaan läksyissä. – They help each other with their homework / while doing homework.

Läksyissä is idiomatic and neutral; läksyissä vs läksyjen kanssa is more about style than grammar. Läksyssä (singular) would sound a bit odd unless you really mean one specific exercise.

Could the word order be different, like Toisiaan he auttavat läksyissä? How does word order change the meaning?

Yes, Finnish word order is relatively flexible. All of these are grammatically correct:

He auttavat toisiaan läksyissä. (neutral)
Toisiaan he auttavat läksyissä. (emphasis on each other)
Läksyissä he auttavat toisiaan. (emphasis on in homework)

The basic information remains the same, but the focus shifts:

Starting with He is the most neutral, subject-focused.
Starting with Toisiaan stresses that the important part is that they help each other (maybe in contrast to helping someone else).
Starting with Läksyissä stresses during homework (maybe contrasting with other situations).

So word order in Finnish signals emphasis and information structure more than grammar.

Is this how people really say it in spoken Finnish, or would it sound different in everyday speech?

In everyday spoken Finnish, the sentence often changes a bit:

Standard written:
He auttavat toisiaan läksyissä.
Typical spoken:
Ne auttaa toisiaan läksyissä.

Main differences:

he → ne for people
auttavat → auttaa (3rd person plural often uses the 3rd singular verb form in speech)

You might also hear some pronunciation simplifications (like läksyis instead of läksyissä), but Ne auttaa toisiaan läksyissä is a good spoken equivalent.

Can you leave out he and just say Auttavat toisiaan läksyissä?

Yes, grammatically you can omit the subject pronoun when it is clear from context:

Auttavat toisiaan läksyissä. – (They) help each other with homework.

Finnish often drops personal pronouns because the verb ending shows the person and number. However:

In written Finnish, you usually include the pronoun (He auttavat…) unless the subject is very obvious from the surrounding text.
In speech, dropping it is more common, especially if you have just mentioned who they are.