Learn Hardcore German: Wenn die Hausverwaltung nicht reagiert, schreibe ich eine zweite Beschwerde. - If the property management doesn’t respond, I write a second complaint.

Because wenn introduces a subordinate clause, and in German subordinate clauses send the conjugated verb to the end.
So you get: Wenn die Hausverwaltung nicht reagiert (verb reagiert at the end).

Why is there a comma, and why is the word order schreibe ich (not ich schreibe)?

A subordinate clause in front of the main clause is followed by a comma:
Wenn … , …

When the subordinate clause comes first, it takes the first position in the sentence. Then the main clause must follow the verb-second (V2) rule, so the finite verb comes next, and the subject comes after it:
Wenn …, schreibe ich …
If you reverse the order, you’d say: Ich schreibe …, wenn …

Is this a real conditional, like if in English?

Yes. Wenn often corresponds to English if/when. In contexts like this (a consequence depending on a condition), it’s typically translated as if. The grammar pattern is a standard conditional:
Wenn + condition, (then) + result.

Why is it Wenn and not Wann?

Wenn is used to introduce a condition (if) or a time clause (when).
Wann is used for direct or indirect questions about time:

Wann reagiert die Hausverwaltung? (When do they respond?)
Ich weiß nicht, wann sie reagiert. (I don’t know when they’ll respond.)

Why is it die Hausverwaltung—what exactly does that mean, and why die?

Hausverwaltung means property management / building management (the office/company managing the building).
It’s grammatically feminine in German, so it takes die in the nominative singular: die Hausverwaltung.

What case is die Hausverwaltung here?

It’s nominative because it’s the subject of the subordinate clause (the thing that does or doesn’t respond):
die Hausverwaltung … reagiert.

Why is nicht placed before reagiert?

In this clause, nicht negates the verb/action reagieren (to respond). In many simple clauses, nicht appears shortly before what it negates, and with a single-verb predicate in a subordinate clause, that often means right before the final verb:
… nicht reagiert = does not respond.

What tense is reagiert / schreibe—and does it refer to the future?

Both are present tense (Präsens):

(sie) reagiert = she/it responds
(ich) schreibe = I write

German often uses Präsens for future meaning when the context makes it clear (here, the condition implies a future possible situation). You could also use werde … schreiben, but it’s not necessary.

Why is it eine zweite Beschwerde and not something like die zweite Beschwerde?

eine zweite Beschwerde means a second complaint (one more complaint, another one).
die zweite Beschwerde would usually mean the second complaint in a specific known sequence (e.g., you already have a numbered list or both speakers know exactly which one is “the second”).

Why does zweite have the ending -e (not zweiten or zweiter)?

Because Beschwerde is feminine, and the phrase uses the indefinite article eine in the accusative. After ein-/eine- words, adjectives typically take the weak ending pattern, and for feminine accusative that’s -e:
eine zweite Beschwerde.

(Also: Beschwerde is the direct object of schreiben here, so accusative is expected.)

Can you add dann (then): …, dann schreibe ich …? Is it more natural?

Yes, you can. It often adds emphasis/clarity:
Wenn die Hausverwaltung nicht reagiert, dann schreibe ich eine zweite Beschwerde.
Without dann is perfectly normal and slightly more neutral/compact.

Is wenn the only option here, or could I use falls?

You can also use falls, which is closer to in case / if it turns out that and can sound a bit more cautious/less certain:

Wenn die Hausverwaltung nicht reagiert, … (if/when they don’t respond)
Falls die Hausverwaltung nicht reagiert, … (in case they don’t respond)

Both are correct; the difference is mainly nuance.

Is the sentence formal enough for writing to property management? Any alternatives?

It’s fine and neutral-formal. If you want it to sound more “official,” you could choose slightly more administrative verbs, e.g.:

… reiche ich eine zweite Beschwerde ein. (I submit a second complaint.)
… werde ich eine zweite Beschwerde einreichen. (more explicitly future)
But your original wording is already appropriate.