Learn Hardcore Japanese: watasitati ha muryou no ibento ni sumaho de mousikomimasita. - We applied for a free event with a smartphone.

は marks the topic: “As for us, …” It sets the background and sounds neutral.
が marks the subject with emphasis/contrast: 私たちが… would stress “we (not someone else) applied.” Both are grammatically fine; は is the most natural here since you’re simply reporting what “we” did.

Why is there の between 無料 and イベント? Why not 無料な or just 無料イベント?

無料 is a noun meaning “no charge,” so to modify another noun you typically use A の B: 無料のイベント.
無料なイベント is unnatural because 無料 isn’t a な-adjective.
無料イベント (compound noun) is also common, especially in headlines/ads. In regular speech, 無料のイベント feels a bit more neutral.

Why is イベント followed by に? Why not を?

With 申し込む, use:

X に申し込む = apply/sign up “to/for X.” Example: 大会に申し込む.
Y を申し込む = request/book Y (the thing you are requesting). Example: 参加を申し込む, ホテルの予約を申し込む. So イベントに申し込む = sign up for the event. Saying イベントを申し込む would sound like you’re “requesting an event,” which isn’t what you mean.

What does で after スマホ mean? Could I use から or を使って?

で marks the means/method: スマホで = “by/using a smartphone.”
スマホから = “from a smartphone” (origin/source). It’s common in casual speech but focuses on the source rather than the means.
スマホを使って explicitly says “using a smartphone.” It’s fine but longer. スマホで is the default and most natural here.

Is the word order fixed? Can I move スマホで earlier?

Word order is flexible as long as particles stay attached. All of these are fine:

私たちはスマホで無料のイベントに申し込みました。
私たちは無料のイベントにスマホで申し込みました。
スマホで、私たちは無料のイベントに申し込みました。 Shifts change nuance/emphasis slightly but not the core meaning.

Is 申し込みました one word? Why not 申し込ました?

Dictionary form: 申し込む.

Polite non-past: 申し込みます (the ます attaches to the -i stem: 申し込み).
Polite past: 申し込みました. Forming 申し込ました is incorrect; you must use the -i stem before ます.

What’s the difference between 申し込む and 申し込み? Is 申し込みをしました okay?

申し込む = verb “to apply/sign up.”
申し込み = noun “application/registration.” Both patterns are fine:
（イベントに）申し込みました。 (short, very common)
（イベントに）申し込みをしました。 (noun + する; a bit heavier/formal) In formal writing you may also see 申し込みを行いました.

Why ました and not でした?

ました attaches to verbs (e.g., 申し込みました).
でした attaches to nouns/な-adjectives (e.g., 無料でした = “It was free”).

Can I drop 私たち? Do I need to say “we” in Japanese?

Yes, you can omit it if context makes it clear:

（私たちは）無料のイベントにスマホで申し込みました。 Japanese often drops pronouns. Alternatives for “we” include 僕ら, 私たち, 私たちみんな, etc. Note 〜たち pluralizes loosely (“and others with me”).

What level of politeness is 申し込みました? How would I say it casually?

申し込みました = polite past.
Casual past: 申し込んだ.
Polite negative past: 申し込みませんでした.
Casual negative past: 申し込まなかった.

Could I use 応募した, 登録した, or 予約した instead of 申し込んだ? What’s the nuance?

申し込む: generic “sign up/apply” (events, services, classes).
応募する: apply for something selective/competitive (jobs, contests, lotteries).
登録する: register (create an account, add your info to a system).
予約する: reserve/book (seats, hotels, restaurants). For a free event sign-up, 申し込む is the safest default.

Why not use へ with 申し込む (イベントへ申し込む)?

へ marks direction with motion verbs (行く, 来る). 申し込む isn’t a motion verb; it takes に to mark the target of the application, so イベントに申し込む is natural.

Does the 申 in 申し込む make it honorific/humble?

No. Although 申す is a humble verb meaning “to say,” the compound 申し込む is a lexicalized neutral verb. It isn’t keigo by itself. For politeness, use polite forms or set phrases like お申し込みください (“please apply”).