Learn Hardcore German: Die erste Beschwerde war höflich, und trotzdem hat der Nachbar im Flur weiter geschimpft. - The first complaint was polite, and still the neighbor male kept complaining in the hallway.

Because Beschwerde is a feminine noun: die Beschwerde.
Adjectives and ordinal numbers (like erste) must match the noun’s gender, case, and number. Here it’s nominative singular feminine, so you get die erste Beschwerde.

What exactly is the grammar of Die erste Beschwerde war höflich?

It’s a simple main clause in past tense using sein:

Subject: Die erste Beschwerde
Verb: war (simple past of sein)
Predicate adjective: höflich
So: subject + verb + adjective.

Why does the second part use hat ... geschimpft instead of schimpfte?

German often prefers the perfect tense (hat ... geschimpft) in spoken and informal narration, while schimpfte (simple past) is more common in written narratives and certain verbs.
Both are correct; hat ... geschimpft is just the more everyday choice.

Why is the verb split: hat ... geschimpft?

That’s the perfect tense structure:

Finite auxiliary in 2nd position: hat
Past participle at the end: geschimpft
In German main clauses, the conjugated verb goes in position 2, and non-finite verb parts (like participles) typically go to the end.

What does trotzdem do to word order, and why is it ... und trotzdem hat ...?

trotzdem is a “conjunct adverb” (a linking adverb). If it starts a clause, it takes the first position, and the finite verb still must be in position 2.
So after trotzdem, you get inversion:

trotzdem (position 1)
hat (position 2)
then the subject: der Nachbar

That’s why it’s trotzdem hat der Nachbar ....

Could I also say ..., trotzdem der Nachbar im Flur weiter geschimpft hat?

Not like that. trotzdem does not introduce a subordinate clause.
If you want a subordinate-clause structure, you’d typically use something like:

obwohl: ..., obwohl der Nachbar im Flur weiter geschimpft hat.
With obwohl, the finite verb goes to the end.

Why is there a comma before und trotzdem?

German uses a comma to separate two main clauses when they are connected and each has its own finite verb: 1) Die erste Beschwerde war höflich
2) trotzdem hat der Nachbar ... geschimpft
Since these are two independent clauses, the comma is standard before und here.

What case is der Nachbar, and why?

der Nachbar is nominative because it’s the subject of the second clause: the person doing the action (schimpfen).
So: hat der Nachbar ... geschimpft = the neighbor complained/scolded.

Why is it im Flur and not in der Flur?

Because im is a contraction of in dem.
Also, Flur is masculine in the sense of hallway/corridor: der Flur.
So:

in dem Flur → im Flur (dative)

It’s dative because in describes a location here (not movement).

What does weiter mean here, and where does it go in the sentence?

weiter means to continue / still / further in the sense of continuing an action: kept on complaining/scolding.
Position-wise, it’s an adverb and sits in the “middle field,” typically before or near the main verb phrase: ... im Flur weiter geschimpft.

What is the difference between schimpfen and beschweren in this context?

(sich) beschweren = to complain (often more neutral/formal; can be polite)
schimpfen = to scold/complain angrily, rant, grumble (more emotional/negative)

That contrast matches the sentence: the complaint was polite, but the neighbor kept ranting.

Does höflich need an ending here (like höfliche)?

No. After linking verbs like sein, adjectives used as predicate adjectives do not take endings:

Die Beschwerde war höflich.
Endings like höfliche are used when the adjective comes before a noun:
eine höfliche Beschwerde.