Wenn die Hausverwaltung nicht reagiert, schreibe ich eine zweite Beschwerde.

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).

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 …

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.

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.)

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.

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

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.

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”).

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.)

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.

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.

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.