La infano trankvile eniras la ĉambron, por ne veki la avinon.

Because trankvile is an adverb, and adverbs in Esperanto usually end in -e. It describes how the child enters.

• trankvile = calmly / quietly
• trankvila = calm, as an adjective

So:

• La infano trankvile eniras... = The child enters calmly/quietly.
• La infano estas trankvila. = The child is calm.

Because eniri is a normal Esperanto verb meaning to enter. The prefix en- adds the idea of in/into to iri (to go).

So:

• iri = to go
• eniri = to go in / to enter

Esperanto often builds verbs this way with prefixes.

Yes. Both are natural, but they are built differently.

• eniras la ĉambron = enters the room
• iras en la ĉambron = goes into the room

The first uses the verb eniri.
The second uses iri plus the preposition en.

In many situations, they mean almost the same thing.

Here ĉambron has -n because it is the direct object of eniras.

In Esperanto, eniri is transitive: you enter something.

• Li eniras la domon. = He enters the house.
• La infano eniras la ĉambron. = The child enters the room.

This is one place where Esperanto may feel different from English, because English uses enter transitively too, but English learners sometimes expect a preposition because of phrases like go into.

Because la avinon is the direct object of veki.

• veki = to wake someone
• veki la avinon = to wake the grandmother

The child is trying not to do the action to the grandmother, so avinon gets the accusative -n.

por + infinitive expresses purpose: in order not to wake.

So:

• por = for / in order to
• veki = to wake
• por ne veki la avinon = in order not to wake the grandmother

This is especially common when the subject is the same as in the main clause. The child enters quietly, and the same child is the one who does not want to wake the grandmother.

Because ne usually goes directly before the word or phrase it negates.

Here it negates the action veki:

• por ne veki la avinon = so as not to wake the grandmother

That placement is the normal and clearest one.

Yes, but por + infinitive is simpler here.

You could make a fuller clause, for example:

• ... por ke ŝi ne veku la avinon.

But that is more explicit and a bit heavier. Since the subject is already understood, Esperanto normally prefers the shorter por ne veki la avinon.

Yes. Esperanto word order is fairly flexible.

For example, these are also possible:

• La infano eniras la ĉambron trankvile...
• Trankvile la infano eniras la ĉambron...

The basic meaning stays the same. Word order mainly changes emphasis or style. Because Esperanto marks objects with -n, the roles in the sentence stay clear.

Because la is the definite article in Esperanto, and it is used for specific things or people.

Here the sentence is talking about specific referents:

• la infano = the child
• la ĉambro = the room
• la avino = the grandmother

Esperanto has no indefinite article like English a/an. So infano by itself could mean a child or just child depending on context.

This is an important Esperanto distinction:

• veki = to wake someone
• vekiĝi = to wake up

So:

• La infano ne volas veki la avinon. = The child does not want to wake the grandmother.
• La avino ne volas vekiĝi. = The grandmother does not want to wake up.

In your sentence, the child is trying not to wake someone else, so veki is correct.

Not always. Many writers would omit it:

• La infano trankvile eniras la ĉambron por ne veki la avinon.

With the comma, the writer is marking a small pause. Without it, the sentence is also perfectly normal. So this is often a matter of style rather than meaning.

No. infano is gender-neutral. It just means child.

If you want to be specific, you can say:

• knabo = boy
• knabino = girl

But infano leaves the gender unspecified, which is very common and natural.