Quiero volver a casa después de la reunión.

In Spanish, subject pronouns are often omitted because the verb ending already tells you who the subject is.

• quiero = I want
• The ending -o shows it is yo

So Quiero volver... is perfectly natural.
You could say Yo quiero volver... if you want extra emphasis, for example to contrast with someone else:

• Yo quiero volver, pero ellos no. = I want to go back, but they don’t.

Because after querer when you say you want to do something, Spanish uses:

querer + infinitive

So:

• Quiero volver = I want to return / go back
• Quiero comer = I want to eat
• Quiero salir = I want to leave

This is very similar to English want + to + verb, except Spanish does not need a separate word for to here.

Here, a is the preposition meaning to.

• volver a casa = to return home
• ir a Madrid = to go to Madrid
• llegar al hotel = to arrive at the hotel

Be careful: volver a + infinitive can also mean to do something again:

• Volver a llamar = to call again

But in your sentence, a casa is a destination, so it means return home, not do home again.

Spanish often uses casa without an article when it means home in a general, personal sense.

• volver a casa = to return home
• estar en casa = to be at home
• llegar a casa = to get home

You would use la casa more when referring to a specific house/building:

• Volví a la casa de Ana. = I went back to Ana’s house.

So a casa is the natural choice for home.

después de means after when it is followed by a noun.

• después de la reunión = after the meeting
• después de la cena = after dinner

A useful rule:

• después de + noun
• después de + infinitive
• después de que + clause

Examples:

• después de clase = after class
• después de comer = after eating
• después de que terminó la reunión = after the meeting ended

Because después normally needs the preposition de before a noun or infinitive.

So these are correct:

• después de la reunión
• después de comer

And this is not standard:

• después la reunión

Think of después de as a fixed expression in this structure.

Yes, reunión is a feminine noun, so it takes la:

• la reunión = the meeting

Many nouns ending in -ión are feminine:

• la conversación
• la decisión
• la estación

Plural:

• la reunión
• las reuniones

Because Spanish only makes contractions in two cases:

• a + el = al
• de + el = del

There is no contraction with la.

So:

• de la reunión = correct
• a la casa = correct
• del hotel = from the hotel
• al centro = to the centre/downtown

Yes. Regresar a casa and volver a casa both mean to return/go back home.

• Quiero volver a casa
• Quiero regresar a casa

Both are correct. In everyday Spanish, volver is extremely common. Depending on region and style, one may sound more natural than the other, but both are widely understood.

Yes. Spanish is fairly flexible with word order, especially with time expressions.

All of these are possible:

• Quiero volver a casa después de la reunión.
• Después de la reunión, quiero volver a casa.
• Quiero, después de la reunión, volver a casa. (less common in neutral speech, but possible)

The original version is the most straightforward and natural.

The stress is on the last syllable: reu-nión.

A rough guide:

• quiero ≈ KYEH-ro
• volver ≈ bol-BEHR
• reunión ≈ reh-oo-NYON

A few pronunciation notes:

• qu before i/e sounds like a hard k
• v and b are pronounced very similarly in Spanish
• the written accent in reunión shows where the stress goes

Yes, it sounds completely natural in Spain.

It is a normal, everyday sentence and uses very standard vocabulary:

• quiero = I want
• volver a casa = return home
• después de la reunión = after the meeting

A speaker in Spain would understand it immediately, and it does not sound translated or unnatural.