Matene mi lavas mian vizaĝon per malvarma akvo, sed ne miajn okulojn.

Matene is an adverb made from mateno (morning).

• mateno = morning
• matene = in the morning / mornings / during the morning

The ending -e makes it adverbial, so it describes when the action happens.

A native English speaker may expect a preposition, like in the morning, but Esperanto often uses a simple adverb instead:

• Matene mi lavas... = In the morning, I wash...

You could also say en la mateno, but matene is shorter and very natural.

Esperanto present tense often covers both ideas.

• mi lavas can mean I wash, I am washing, or I do wash
• the exact meaning depends on context

Here, because of Matene and the general, routine-like feel of the sentence, it most naturally means something like:

• In the morning I wash my face...

So it sounds like a habit or regular action, not necessarily something happening right this second.

Because mian vizaĝon is a noun phrase in which the words must agree.

• vizaĝo = face
• vizaĝon = face as a direct object
• mia = my
• mian = my, agreeing with a singular accusative noun

In Esperanto, adjective-like words agree with the noun they describe in:

• number
• case

So:

• mia vizaĝo = my face
• mian vizaĝon = my face as the direct object

This is very normal Esperanto grammar.

That -n marks the direct object.

In this sentence, the verb is lavas (wash), and the things being washed are:

• mian vizaĝon = my face
• miajn okulojn = my eyes

So they take the accusative ending -n.

English usually shows this by word order alone, but Esperanto marks it directly on the object.

Because okuloj means eyes, and that is plural.

The full breakdown is:

• okulo = eye
• okuloj = eyes
• mia = my
• miaj = my, plural
• miajn okulojn = my eyes, as a direct object

So both words show plural, and both also show the accusative:

• plural: -j
• accusative: -n

That gives:

• miajn okulojn

Because per is a preposition, and after an ordinary preposition Esperanto normally does not use the accusative -n.

Here:

• per = with / by means of
• malvarma akvo = cold water

So:

• per malvarma akvo = with cold water

Also, malvarma agrees with akvo, which is singular and not accusative here.

So the basic form is correct:

• per malvarma akvo

This is an example of ellipsis: Esperanto leaves out words that are understood from the previous clause.

The full idea is:

• sed mi ne lavas miajn okulojn

But Esperanto can shorten that to:

• sed ne miajn okulojn

So the sentence means:

• ...but not my eyes

The verb is understood from the first part.

This is why miajn okulojn still stays in the accusative: it is still the direct object of the omitted verb lavas.

Because miajn okulojn is still the object of the understood verb.

Compare:

• mi lavas miajn okulojn = I wash my eyes

So in the shortened contrast:

• sed ne miajn okulojn = but not my eyes

If you said miaj okuloj, that would be nominative, which is the form normally used for a subject, not for a direct object. That would not fit the grammar here.

Yes, often it could.

With body parts, Esperanto frequently uses la when the owner is already obvious from the subject:

• Mi lavas la vizaĝon = I wash my face

That is very natural Esperanto.

Using mian vizaĝon is also correct. It is a bit more explicit, and it can feel slightly more emphatic or simply more straightforward for a learner.

So both are possible, but:

• la vizaĝon is often more idiomatic in contexts where the possessor is obvious
• mian vizaĝon is still completely correct

Because sia/sian/siajn is the reflexive possessive used only for a third-person subject.

So:

• Li lavas sian vizaĝon = He washes his own face
• Ŝi fermis siajn okulojn = She closed her own eyes

But with mi, you use mia/mian/miajn, not sia.

So here:

• mi ... mian vizaĝon
• mi ... miajn okulojn

are correct.

The word order is fairly flexible.

This sentence begins with Matene to set the time right away:

• Matene mi lavas mian vizaĝon...

But you could also say:

• Mi lavas mian vizaĝon matene...

Both are grammatical.

Putting Matene first gives it a slight topic-setting feel, like:

• In the morning, ...

That is very common and natural in Esperanto.