Learn Hardcore Russian: Вечером я хочу поработать дома. - In the evening I want to work at home for a while.

So вечером literally is the instrumental form of вечер (evening), but it functions adverbially and already means “in the evening”.
Saying в вечер for “in the evening” is ungrammatical in standard Russian.

What case is вечером, and what does that case do here?

Вечером is in the instrumental case (singular: вечер → вечером).

Here, the instrumental is used in a very common time expression meaning “at / in [that time of day]”. So the instrumental case here:

does not mean “with” or “by” (its other common function),
but instead marks time when something happens: “in the evening”.

Can I move вечером to the end and say Я хочу поработать дома вечером? Does it change the meaning?

Yes, you can say:

Вечером я хочу поработать дома.
Я хочу поработать дома вечером.

Both are correct and both mean roughly “In the evening I want to work at home.”

The difference is mostly about emphasis / focus:

Вечером я хочу поработать дома.
Puts a bit more emphasis on “in the evening” (as opposed to some other time).
Я хочу поработать дома вечером.
More neutral, the whole thing feels like one block: “I want to work at home in the evening.”

In everyday speech, both are natural; the nuance is subtle.

What is the difference between поработать and работать?

Both come from the verb работать (to work), but:

работать – imperfective: to work (general, ongoing, process)
- Я хочу работать дома. – I want to work at home (in general, as a habit or job).
поработать – perfective with the prefix по-, here meaning “to work for a while / to do some work”.
- Я хочу поработать дома. – I want to do some work at home (for a certain time, get some work done).

So поработать often suggests a limited period, “to work a bit / for some time” or “to get some work done (as a one-time action)”.

Why is поработать considered perfective if it looks like an infinitive?

Aspect (perfective vs imperfective) is independent of tense and also applies to infinitives.

работать – imperfective infinitive
поработать – perfective infinitive

Perfective infinitives usually imply:

a single, complete action in the future from the speaker’s point of view, or
the result or boundedness of the action.

So я хочу поработать implies “I want to (sit down and) do some work (and be done with it)”, not “I want to be working (as a general ongoing state)”.

Why is it дома and not в доме?

Both exist but mean different things:

дома – “at home” (no preposition; functions like an adverb of place)
- Я хочу поработать дома. – I want to work at home.
в доме – “in the house / inside the house (building)”
- Я живу в доме напротив. – I live in the house opposite (in that building).

In your sentence, you mean “at home (not at the office, café, etc.)”, so дома is the natural choice.

What’s the difference between дома, домой, and из дома?

These are classic location/direction distinctions:

дома – “at home” (location, where?)
- Я дома. – I am at home.
домой – “(to) home” (direction, where to?)
- Я иду домой. – I’m going home.
из дома – “from home” (direction from where?)
- Я вышел из дома. – I went out from home / left the house.

In your sentence поработать дома, дома answers “where?” (where do I want to work?) → “at home”.

Why is хочу in the present tense if this is about the future evening?

In Russian, verbs of desire or intention in the present tense plus an infinitive often express a present intention about a future action:

Я хочу поработать вечером. – Right now, I have the desire/intention to do some work in the evening.

So:

хочу – present: “I want (now)”
поработать вечером – the action you intend to do in the future.

This is similar to English “I want to work at home this evening”: the verb “want” is present, but the working is in the future.

With хочу поработать, why is the second verb (поработать) not conjugated?

In Russian, verbs that express desire, ability, necessity, etc., are typically followed by an infinitive, not a conjugated verb. For example:

Я хочу поработать. – I want to work.
Я могу поработать. – I can work.
Я должен поработать. – I must work.

So хочу is conjugated (1st person singular), and поработать stays in the infinitive form. This is the standard pattern: modal/mental verb + infinitive.

Could I say Я хочу работать дома вечером instead? How is that different?

Yes, that’s also correct, but the nuance changes:

Я хочу поработать дома вечером.
= I want to do some work at home this evening (one-time action, get some work done).
Я хочу работать дома вечером.
= I want to work at home in the evenings / I want my working time to be in the evenings at home (sounds more like a general preference or plan, not necessarily just tonight).

So поработать suggests a more concrete, one-off action, while работать can sound more habitual or general in this context.

What is the difference between Я хочу поработать and Мне хочется поработать?

Both can be translated as “I feel like working / I want to work”, but the nuance is:

Я хочу поработать дома.
- More direct, volitional: “I want to work at home.”
- Neutral, straightforward statement of will.
Мне хочется поработать дома.
- Literally: “It is wanted by me to work at home.”
- Sounds more like a spontaneous desire, mood, or vague feeling: “I feel like doing some work at home.”

In everyday speech:

я хочу… – stronger, more intentional.
мне хочется… – softer, more like a passing wish or mood.

How would I say “In the evenings I like to work at home” using a similar structure?

For a habitual action in multiple evenings, you’d typically use по вечерам (“in the evenings”) and the imperfective работать:

По вечерам я люблю работать дома.
= In the evenings I like to work at home.

Here:

по вечерам – “in the evenings” (repeated, habitual time)
люблю работать – “I like to work” (general preference)
дома – “at home”.