Learn Hardcore Russian: После совещания мне хочется поехать домой и отдохнуть. - After the meeting, I feel like going home and resting.

The preposition после (after) requires the genitive.
So совещание (nominative) changes to совещания (genitive) to mean after the meeting.
Pattern: после + Genitive → после урока, после работы, после совещания.

Why does the sentence use мне хочется instead of я хочу?

Мне хочется is an impersonal structure meaning something like I feel like / I have the desire to (often more spontaneous, mood-based, or softer).
Я хочу is more direct and deliberate: I want.
Grammar-wise:

мне is dative (literally: to me)
хочется is a 3rd-person singular form used impersonally (no explicit subject like я)

Both can be correct, but мне хочется often sounds more natural for “I feel like doing X (right now/after something).”

What case is мне, and why is it in that case?

Мне is the dative form of я.
In the construction кому хочется (who feels like / who wants), the “person experiencing the desire” is put in the dative:

мне хочется = (lit.) to me it is wanted / it feels wanted
тебе хочется, ему хочется, нам хочется, etc.

What is хочется exactly—what verb form is it?

Хочется is the 3rd person singular present form of хотеться (a reflexive verb). It’s commonly used impersonally:

Мне хочется спать = I feel like sleeping.
Ей хочется домой = She wants to go home (feels like going home).

The -ся is part of the verb and is normal here.

Why does поехать mean “to go” here, and how is it different from ехать?

Поехать is perfective and often means to set off / to go (by transport) as a single completed departure.
Ехать is imperfective, focusing on the process of traveling.

In “After the meeting, I feel like going home,” Russian often uses the perfective поехать to express the idea of going off (starting the trip):

хочется поехать домой = feel like heading home
хочется ехать домой can sound more like “feel like being on the road/traveling home (the process).”

Does поехать imply using a vehicle?

Usually, yes: поехать is most natural when you mean going by some form of transport (car, bus, metro, etc.).
If you specifically mean on foot, you’d typically use пойти:

хочется пойти домой = feel like going home (walking)

That said, in everyday speech поехать домой can still be used even if the exact transport isn’t important—just the idea of “heading home.”

Why is домой not in a case like в дом or в доме?

Домой is an adverb meaning (to) home—it already contains the idea of direction, so you don’t need a preposition.
Compare:

домой = (go) home (direction)
дома = at home (location)
в дом = into the house (direction into a building)
в доме = in the house (location)

Here, поехать домой is the standard way to say to go home.

Why are there two infinitives: поехать and отдохнуть?

Because хочется can govern an infinitive (or several infinitives) describing what you feel like doing.
Here the speaker feels like doing two actions: 1) поехать домой (go home)
2) отдохнуть (have a rest)

Russian often chains infinitives like this after a verb of desire/state: хочется + infinitive(s).

Why is отдохнуть perfective, and could it be отдыхать instead?

Отдохнуть is perfective: it means to rest (and get some rest as a result)—a complete “resting event.”
Отдыхать is imperfective: it means to be resting / to rest (as a process or habit).

In this context (“after the meeting I feel like going home and resting”), отдохнуть is very natural because the speaker wants to get some rest (a completed, satisfying result).

Is there any punctuation needed before и in поехать домой и отдохнуть?

No comma is needed here because поехать and отдохнуть are two infinitives connected by и and share the same governing word хочется.
A comma would appear if you had separate clauses, but here it’s one simple structure: хочется (что сделать?) поехать… и отдохнуть.

Can the word order be changed, and what would it change?

Yes—Russian word order is flexible, but it affects emphasis. For example:

После совещания мне хочется поехать домой и отдохнуть. (neutral)
Мне после совещания хочется поехать домой и отдохнуть. (emphasizes to me / personally)
Домой мне хочется поехать после совещания и отдохнуть. (emphasizes home as the destination)

The original order is very standard and natural.

How would you say the same idea more formally or more casually?

Common variants:

More direct: После совещания я хочу поехать домой и отдохнуть. (clear, straightforward)
A bit more formal/neutral: После совещания мне хотелось бы поехать домой и отдохнуть. (I would like to… softer, polite)
More conversational: После совещания так хочется домой, отдохнуть. (very natural spoken style; can drop the verb поехать because домой already implies “go home”)