Learn Hardcore Japanese: kare ha matigaete hitotu mae no eki de orite simai, tikamiti wo sagasita. - He got off one station early by mistake and looked for a shortcut.

It adds a sense of completion plus unintendedness/regret. Here it reads as “he ended up getting off (by mistake),” matching 間違えて. Without 〜てしまい (just 降りて), it would be a more neutral “got off” without the “oops” nuance.

Can I write 降りてしまって instead of 降りてしまい? What’s the difference?

Yes. Both are correct.

降りてしまって is the common conversational connector (“…and then…”).
降りてしまい、 uses the continuative (ます-stem) しまい to link to the next clause; it’s a bit more written/literary. The meaning is the same.

Why is it 駅で降りる and not 駅に降りる?

Because で marks the place where an action occurs. You get off “at” a station: 駅で降りる.
Compare:

駅に着く (arrive at a station) uses に for destination.
With vehicles, you can also say 電車を降りる (get off the train) or バスから降りる (get off from a bus), but not 駅を降りる.

Why 間違えて and not 間違って? Are both OK?

Both appear in real usage, but they come from different verbs:

間違えて (from 間違える “to make a mistake”) highlights the doer making a mistake. Very natural for “by mistake.”
間違って (from 間違う “to be wrong”) can mean “wrong/incorrect” or “by mistake” in casual speech. In this sentence either works; 間違えて slightly emphasizes the speaker’s mistaken action.
Examples:
間違えて送信した。 I sent it by mistake.
番号が間違っている。 The number is wrong.

What exactly does 一つ前の駅 mean? Are there other natural ways to say it?

It means “the station one before (the intended one).” Reading: ひとつまえのえき.
Natural alternatives:

一つ手前の駅 (ひとつてまえのえき) — very common; “one station short of [the destination].”
一駅手前で (ひとえきてまえで) — also common, compact. All are fine; 手前 emphasizes “short of (your goal).”

What is の doing in 一つ前の駅?

の links the modifier to the noun: [(一つ前) の (駅)]. It’s “the station that is one-before.” It’s the standard way to attach a descriptive phrase to a noun.

Why 彼は and not 彼が? And why isn’t 彼 repeated after the comma?

は marks the topic (“as for him”), natural in narrative.
が would put focus on who did it (“it was he who…”), useful for contrast or new information.
Japanese drops repeated subjects when clear from context, so after the comma it stays understood as “he.”

Which kanji for おりる is correct here: 降りる or 下りる?

Use 降りる for getting off vehicles. 下りる is typically for going down from a higher place (stairs, slopes). Many writers simply use kana (おりる) to avoid the kanji choice, but here 降りる is best.

Does 近道を探した mean he found a shortcut?

No. 探す means “to look for,” not necessarily find. If he found one, you’d say 近道を見つけた. If you want “tried to look for,” use 近道を探してみた.

Is the politeness consistent? How would I say this politely?

Yes—politeness is set by the final verb. To make it polite:

彼は間違えて一つ前の駅で降りてしまい、近道を探しました。 Using 降りてしまい is fine in both plain and polite sentences.

Are the spaces normal in Japanese writing?

No. Standard Japanese doesn’t use spaces between words. They’re included here for learners. In ordinary text it would be written without spaces: 彼は間違えて一つ前の駅で降りてしまい、近道を探した。

Can I move 間違えて to another position, like 一つ前の駅で間違えて降りてしまい?

Yes. 間違えて is an adverbial (“by mistake”) modifying 降りてしまい, so both

間違えて一つ前の駅で降りてしまい
一つ前の駅で間違えて降りてしまい are natural. The meaning is the same; it’s just a slight difference in rhythm/emphasis.