Learn Hardcore Chinese: wǒ zài jiā xiě zhōngwén zuòyè.

You can think of 在 here as a verb meaning “to be located at” (at/in). So the whole thing is more like “I am at home (and) write Chinese homework.” There’s no extra copula needed.

What exactly does 在 mean here? Is it “at,” “in,” or something like an -ing marker?

Here 在 is mainly a location marker:

在 + place = at / in + place

So 在家 = at home.

There is another use where 在 (or 正在) comes before a verb to show a progressive action:

我在写中文作业。 = I am writing Chinese homework (right now).

In your sentence, 在 is tied to 家 (place), not directly to 写. If you really want to emphasize the ongoing action, you can say:

我在家正在写中文作业。 = I am (currently) at home, writing Chinese homework.

Why is it 在家 and not 在我的家 or 在家里?

All three are possible, but they feel different:

在家
- Default, very natural.
- Usually implies “at (my) home” if the subject is 我.
- Most common in everyday speech.
在家里
- Literally “inside the home/house.”
- Slight extra emphasis on inside the home, but often interchangeable with 在家.
在我的家
- Emphasizes “at my home (not someone else’s)”.
- Used when you need to contrast locations, e.g.
  - 他在他家，我在我的家。 = He’s at his home; I’m at my home.
- Sounds a bit heavy if there’s no need for that contrast.

For a simple statement like this, 我在家写中文作业 is the most natural.

Why is the order 我在家写中文作业 and not something like 我写在家中文作业?

Mandarin has a fairly stable word-order pattern:

Subject + (Time) + 在 + Place + Verb + Object

So:

我 (subject)
在家 (place phrase with 在)
写 (verb)
中文作业 (object)

You generally cannot put 在 + place between the verb and its object in this kind of sentence, so:

✗ 我写在家中文作业 — unnatural / wrong
✓ 我在家写中文作业 — correct

If you add a time word, it fits like this:

我今天在家写中文作业。
(Subject + Time + 在 + Place + Verb + Object)

How do I know if it means “I am writing” or “I write” (habitually)? There’s no tense.

Mandarin verbs usually don’t change form for tense. 我在家写中文作业 can mean:

“I am writing Chinese homework at home (now).”
“I (usually) write my Chinese homework at home.”

Context decides which reading is intended. To make it clearer:

For ongoing / right now:
- 我在家正在写中文作业。
- 我在家写中文作业呢。
For habit / routine:
- 我常常在家写中文作业。
- 我一般在家写中文作业。

Can I say 做中文作业 instead of 写中文作业?

Yes, and they’re both common, but there’s a nuance:

做作业 = do homework (general; any type of tasks)
写作业 = write homework (emphasizes the writing part)

For 中文作业, which often involves writing characters, both are possible:

我在家做中文作业。 = I do my Chinese homework at home.
我在家写中文作业。 = I (specifically) write my Chinese homework at home.

If the homework is mainly writing characters, 写 feels very natural. If it also includes reading, listening, exercises, etc., 做中文作业 sounds more general.

What’s the difference between 中文 and 汉语 here? Could I say 汉语作业?

Both 中文 and 汉语 relate to Chinese, but:

中文
- Literally “Chinese writing / Chinese language (as a whole)”.
- Often used for the subject “Chinese” in school, especially including reading and writing.
- 中文作业 sounds very natural as “Chinese class homework.”
汉语
- More specifically the Chinese language of the Han people, usually as a spoken / language system concept.
- 汉语作业 is also understandable, but is heard less often in everyday school talk; textbooks and formal contexts might use it more.

In this context, 中文作业 is the most typical and natural way to say “Chinese homework.”

Why isn’t there a measure word between 写 and 中文作业? Should it be 写一个中文作业?

In Mandarin, you use measure words when you’re counting or specifying quantity:

一份中文作业 = one piece/set of Chinese homework
两项作业 = two assignments/tasks

But when you are just talking about homework in general as an activity, you don’t need a measure word:

写中文作业 = write/do Chinese homework (in general)

Saying 写一个中文作业 sounds odd, because 作业 is usually not treated as a single, countable item in that way. Use a more natural measure word if needed:

写一份中文作业。
完成这份中文作业。

Can I just say 写作业 instead of 写中文作业?

Yes:

写作业 = write/do homework (no subject specified)
写中文作业 = write/do Chinese homework (subject specified)

So:

我在家写作业。 = I do homework at home.
我在家写中文作业。 = I do my Chinese homework at home.

If the context already makes it clear that you’re talking about Chinese homework, people will often just say 写作业.

Can the subject 我 be dropped? For example, just 在家写中文作业。

Yes, if the subject is clear from context, Mandarin often drops it:

(我) 在家写中文作业。

In a conversation where everyone knows you’re talking about your own schedule, 在家写中文作业 would typically be understood as “(I) do my Chinese homework at home.”

Can I move 在家 to other positions, like 在家我写中文作业 or 我写中文作业在家?

在家我写中文作业。
- Possible, but sounds a bit marked/emphatic, like stressing “At home, I write Chinese homework.”
- Often used in contrasts:
  - 在学校我上课，在家我写中文作业。
我写中文作业在家。
- Generally unnatural.
- With this kind of verb and object, the place phrase with 在 normally goes before the verb, not after the object.

The safest and most natural pattern is:

我在家写中文作业。
我今天在家写中文作业。
放学以后我在家写中文作业。

Is 我在家写中文作业 formal, informal, or neutral?

It’s neutral and everyday:

Very natural in spoken Chinese.
Fine in informal writing (messages, diaries, etc.).

In more formal or written contexts (like a report or essay), you might see slightly more formal verbs or structures:

我在家完成中文作业。 = I complete my Chinese homework at home.
我通常在家完成中文作业。

But for normal conversation, 我在家写中文作业 is perfectly appropriate and idiomatic.