Learn Hardcore Japanese: urusaku site simatte, tonari no hito ni ayamaru tumori da. - I was noisy, so I intend to apologize to the neighbor.

Because うるさい is an i-adjective. When i-adjectives modify a verb like する, they take the adverbial -く form: うるさくする (to act in a loud/noisy way; to make things noisy). So うるさくして is the て-form of うるさくする.

i-adjective pattern: 速い → 速くする
na-adjective pattern: 静か（な）→ 静かにする

What exactly is してしまって? Why are there “two て’s”?

It’s stacked morphology:

Base verb: うるさくする (be/make loud)
Add the auxiliary 〜てしまう: うるさくしてしまう (to end up doing; to do completely)
Put that auxiliary into て-form to connect to the next clause: うるさくしてしまって So the sequence is: うるさく + して + しまって.

What nuance does 〜てしまう add here?

〜てしまう often implies completion with an unintended or regrettable feeling. Here, with an apology following, it signals “I ended up being loud (unfortunately).” In casual speech, it often contracts to 〜ちゃう; in this sentence that would be うるさくしちゃって.

Neutral completion: 宿題をやってしまった (I finished my homework)
Regret/unintended: 寝過ごしてしまった (I overslept by accident)

Does the 〜て here mean “because”? Could I use から or ので instead?

Yes. The 〜て link can imply a natural cause/result: “(I ended up being loud,) so…”. You can also make the reason explicit:

うるさくしてしまったので、隣の人に謝るつもりだ。 (more formal, matter-of-fact)
うるさくしてしまったから、隣の人に謝るつもりだ。 (more direct/casual) Using 〜て is slightly softer and more narrative.

What does the の do in 隣の人?

の lets the noun 隣（となり） modify 人: “the person next to/next door,” i.e., the neighbor. Other examples:

隣の席 (the seat next to me)
隣の家 (the house next door) You’ll also hear polite/casual neighbor words like お隣さん or お隣の方.

Why is に used after 隣の人 with 謝る? Could I use を or へ?

With 謝る (to apologize), the person you apologize to takes に: 〜に謝る. Using を is ungrammatical here. へ indicates direction and sounds odd with 謝る. A more formal alternative is 〜に対して謝る.

店員に謝った (I apologized to the clerk)
彼に対して謝る (apologize to him; formal)

What does つもりだ add, and how does it differ from similar expressions?

つもりだ expresses the speaker’s intention/plan.

謝るつもりだ: I intend/plan to apologize (firm personal intention)
謝ろうと思う: I think I’ll apologize (considering it; slightly softer or just-decided)
謝りたい: I want to apologize (desire)
謝る予定だ: I’m scheduled/planning (on the calendar) to apologize (can sound odd unless there’s a set arrangement) Negative: 謝るつもりはない (I don’t intend to apologize).

Is the sentence polite enough? How can I make it more polite?

As written it’s plain style. For polite speech:

うるさくしてしまったので、お隣の方に謝るつもりです。
うるさくしてしまいましたので、お隣の方に謝ろうと思っています。 Using です／ます, and お隣の方 (polite) raises the register.

Do I need to say “I” explicitly?

No. Japanese commonly omits the subject when it’s clear from context. If you need to be explicit, add 私は／僕は:

私は隣の人に謝るつもりだ。

Could I say うるさくなって instead of うるさくして? What’s the difference between なる and する here?

うるさくする: you/your actions make it noisy (agentive; you take responsibility).
うるさくなる: it becomes noisy (state change; may sound less directly your fault). So うるさくなってしまって shifts focus away from your causing the noise.

How could I turn this into a direct apology instead of stating an intention?

Say the apology itself:

うるさくしてしまって、すみません。
More formal: うるさくしてしまい、申し訳ありません。 You can add timing/context: 先ほどはうるさくしてしまって、失礼しました。

Are there common synonyms or set phrases for “being noisy” or “causing trouble” here?

Yes:

騒ぐ → 騒いでしまって (I made a racket)
音を立てる → 音を立ててしまって (I made noise/sounds)
ご迷惑をかける → ご迷惑をおかけして (I caused you trouble/inconvenience; very common in apologies) Examples:
騒いでしまって、隣の人に謝るつもりだ。
ご迷惑をおかけしてしまい、隣の方に謝ろうと思っています。

Any pitfalls with 謝る? What are the readings I should know?

謝る（あやまる） means “to apologize.” Don’t confuse it with 誤る（あやまる）, which means “to make a mistake.”
Formal noun: 謝罪（しゃざい） (apology), e.g., 謝罪する (to make an apology; formal).
Readings: 隣（となり）, 謝る（あやまる）, うるさい (commonly written in kana).