Learn Hardcore German: Danach bin ich schnell zum Balkon gegangen, um den stillen Baum im Licht des Mondes zu sehen. - Afterwards I quickly went to the balcony to see the quiet tree in the light of the moon.

In German, many intransitive verbs that express a change of location or state use sein as their auxiliary in the perfect tense rather than haben.

Gehen (to go) is a movement verb, so it pairs with sein.
Hence you say ich bin gegangen (I have gone), not ich habe gegangen.

Why does the sentence start with Danach, and how does that change the word order compared to Ich bin danach…?

Danach is a sentence‑adverb meaning “after that.” In German main clauses the finite verb must occupy the second position.

By placing Danach first, the auxiliary bin comes directly after it: Danach bin ich…
Alternatively you could start with the subject: Ich bin danach…, but then danach moves to a later slot.

Why is schnell placed before the prepositional phrase zum Balkon gegangen, and is it an adverb or adjective here?

Here schnell functions as an adverb of manner (“quickly”) modifying gegangen. In perfect tense with sein, the usual order is:
Subject + auxiliary + adverb of manner + past participle.
Because Danach is fronted, the structure becomes:
Danach (1) + bin (2) + ich + schnell + zum Balkon gegangen.

What is the case and meaning of zum Balkon?

zu is a dative preposition, so dem Balkon (dative) becomes zum Balkon by contraction (zu dem → zum).
It indicates direction: “to/toward the balcony.”

What function does the um … zu construction have in um den stillen Baum zu sehen?

The um … zu structure expresses purpose (“in order to”). Its pattern is:
um + [accusative object] + zu + [infinitive verb].
So um den stillen Baum zu sehen means “in order to see the silent tree.”

Why does stillen end with ‑n in den stillen Baum?

Baum is masculine and in the accusative case (direct object).
With a definite article (der → den) the adjective takes the weak declension ending ‑en.
Thus: den (article) + stillen (adj. weak declension) + Baum.

What does im Licht des Mondes mean, and why is des Mondes in the genitive case?

im is a contraction of in dem, where in
- dative expresses location: “in the light.”
des Mondes is the genitive singular of der Mond (masculine), showing possession: “of the moon.”
Together: im Licht des Mondes = “in the light of the moon.”

What nuance does the adjective still add to Baum, and are there synonyms in German?

still emphasizes absolute quietness or lack of movement, giving a poetic, almost eerie atmosphere.
Possible synonyms (with slight differences):

ruhig (calm)
leise (quietly, softly)
stumm (mute, but usually for living beings)
Using stiller Baum evokes a tree standing silently in the night.