Paul et Marie veulent tous les deux travailler à l'étranger plus tard.

Because the subject is Paul et Marie, which is they in English.

• Paul = he
• Marie = she
• Paul et Marie = they

So the verb vouloir has to be in the 3rd person plural form:

• je veux
• tu veux
• il / elle veut
• ils / elles veulent

That is why the sentence uses veulent.

Because after vouloir (to want), French normally uses another verb in the infinitive.

So the structure is:

• vouloir + infinitive

Examples:

• Je veux manger = I want to eat
• Nous voulons partir = We want to leave
• Paul et Marie veulent travailler = Paul and Marie want to work

So travailler stays in its basic form because it follows veulent.

Tous les deux means both or more literally the two of them.

In this sentence, it emphasizes that Paul and Marie both want to do this.

So:

• Paul et Marie veulent travailler... = Paul and Marie want to work...
• Paul et Marie veulent tous les deux travailler... = Paul and Marie both want to work...

It adds emphasis that this is true for each of them, not just for them as a pair.

Because tous agrees with a plural idea: les deux = the two.

Here, Paul et Marie refers to two people, so French uses tous.

• tout = singular
• tous = plural masculine or mixed group
• toutes = plural feminine

Since Paul et Marie is a mixed group, French uses the masculine plural form by default:

• tous les deux

If you were talking about two women only, you might also hear or see toutes les deux.

In French, expressions like tous les deux often come after the conjugated verb when they emphasize the subject.

So:

• Paul et Marie veulent tous les deux travailler...

This is very natural.

You can also hear:

• Paul et Marie veulent travailler tous les deux...

That is also possible, but the emphasis shifts slightly.
With veulent tous les deux, the focus is more on the fact that both of them want it.

À l'étranger is a fixed expression meaning abroad or in a foreign country / foreign countries.

It is very common in French:

• vivre à l'étranger = to live abroad
• travailler à l'étranger = to work abroad
• partir à l'étranger = to go abroad

Even though English uses one word, abroad, French often uses this expression with à.

Because à l'étranger is an established expression.

Here, étranger is being used like a noun, meaning something like the foreign world or foreign countries, so it takes the article:

• l'étranger

And because it begins with a vowel sound, le becomes l':

• à + l'étranger

So:

• étranger by itself is often an adjective: un pays étranger = a foreign country
• à l'étranger is the set phrase meaning abroad

Because le becomes l' before a vowel or mute h.

So:

• le + étranger becomes l'étranger

This is called elision.

Other examples:

• le ami → l'ami
• le hôtel → l'hôtel

So à l'étranger is simply the normal contracted form before a vowel sound.

Plus tard means later or later on.

Placed at the end, it sounds very natural in French:

• ... travailler à l'étranger plus tard

This is similar to how English often puts later or later on near the end.

French word order is often flexible with time expressions, but the end of the sentence is a very common place for them.

Yes, sometimes, but the version in the sentence is the most natural and neutral.

For example, you may hear:

• Plus tard, Paul et Marie veulent tous les deux travailler à l'étranger.

But that sounds more like Later, Paul and Marie both want to work abroad, which is less natural if you mean in the future.

For the idea of a future plan or intention, plus tard at the end works best:

• Paul et Marie veulent tous les deux travailler à l'étranger plus tard.

Yes, very much like in English.

In this sentence, veulent works like want to in English:

• They want to work

So vouloir is followed directly by another verb in the infinitive, without any extra word between them.

Compare:

• They want to work
• Ils veulent travailler

French does not put a separate word for to before travailler here.

Yes, this is a very common learner question. Veulent is pronounced roughly like vuhl.

Important points:

• the ending -ent in the ils / elles form is usually not pronounced
• so veulent does not sound like it has an extra syllable at the end

Very roughly:

• veut = vuh
• veulent = also roughly vuhl / vuh

This is one reason spoken French can feel tricky: different spellings may sound very similar.

It is grammatically understandable, but it is not the most natural order.

The original sentence is smoother:

• Paul et Marie veulent tous les deux travailler à l'étranger plus tard.

Why?

• tous les deux naturally stays close to the idea it emphasizes: veulent
• plus tard fits well at the end

French often prefers this kind of rhythm and balance. So your version would probably be understood, but native speakers would usually choose the original wording.