Αυτός θέλει νερό και μετά τσάι.

Yes. Greek verbs usually already show the person, so θέλει by itself can mean he/she/it wants.

That means Αυτός is often added for one of these reasons:

• emphasis: he wants water
• contrast: he wants water, not someone else
• clarity: to make the subject extra clear

So this sentence could also be just Θέλει νερό και μετά τσάι if the context already makes the subject obvious.

Here, Αυτός most naturally means he.

Literally, αυτός can also mean this one or this man/person, depending on context. But in a simple sentence like this, learners will usually understand it as he.

So:

• Αυτός θέλει... = He wants...

Θέλει is the 3rd person singular present tense of θέλω (to want).

So:

• θέλω = I want
• θέλεις = you want
• θέλει = he/she/it wants

In this sentence, it agrees with Αυτός, so it means he wants.

In Greek, just like in English, drinks and other mass nouns often appear without an article when you mean some of that thing in a general sense.

So:

• νερό = water / some water
• τσάι = tea / some tea

If you add the article, you get:

• το νερό = the water
• το τσάι = the tea

In this sentence, no article sounds natural because the meaning is something like He wants water and then tea or He wants some water and then some tea.

Because Greek often leaves out words that are easy to understand from context.

So:

• Αυτός θέλει νερό και μετά τσάι.

is a shorter, natural way of saying:

• Αυτός θέλει νερό και μετά θέλει τσάι.

The second θέλει is understood, so Greek can omit it.

This is very common.

Here, μετά means then, after that, or later.

So:

• νερό και μετά τσάι = water and then tea

It shows sequence: first water, then tea.

Depending on context, μετά can also mean after, but in this sentence then/after that is the best translation.

Because they do different jobs:

• και = and
• μετά = then / after that

So και μετά means and then.

If you removed και, the sentence could still work in some contexts, but και μετά is very natural when you are connecting one thing to the next in sequence.

They are the direct objects of θέλει, so grammatically they are in the accusative case.

However, both words are neuter singular nouns, and in this form the nominative and accusative look the same.

So you do not see a visible change here:

• νερό
• τσάι

That is normal.

A simple approximate pronunciation is:

• Αυτός = av-TOS
• θέλει = THEH-lee
• νερό = neh-RO
• και = keh
• μετά = meh-TA
• τσάι = TSA-ee

So the whole sentence is roughly:

av-TOS THEH-lee neh-RO keh meh-TA TSA-ee

A few notes:

• θ sounds like the th in think
• τσ sounds like ts in cats
• the accent mark shows which syllable is stressed

Yes, Greek word order is more flexible than English word order, but changing it can change the emphasis or make the sentence sound less neutral.

Your sentence:

• Αυτός θέλει νερό και μετά τσάι.

is a natural, straightforward way to say it.

For example, you could also say:

• Θέλει νερό και μετά τσάι.
  Same basic meaning, without the explicit he.

• Αυτός θέλει πρώτα νερό και μετά τσάι.
  This means He wants water first and then tea, with the sequence made even clearer.

So yes, word order can vary, but the original sentence is perfectly normal.

By itself, it can mean either one, depending on context.

It could mean:

• right now: he wants water first and then tea
• in a given situation: that is what he is asking for
• sometimes even a habitual pattern, if the context supports it

Greek present tense often covers both simple present and present-in-context meanings, so context tells you which one is intended.