Learn Hardcore Japanese: muryousyuuri ni tuite tenin ni kikimasita.

The first に is part of the set phrase Noun + について meaning “about/regarding.” So 無料修理について = “about free repairs.”
The second に marks the target person of the action: 店員に聞きました = “asked the store clerk.”

Can I change the word order?

Yes. Japanese word order is flexible as long as particles are correct. All of these are natural:

店員に無料修理について聞きました。
無料修理について店員に聞きました。 (your version)
You can even add a topic: 私は店員に無料修理について聞きました。

Why not use を after 店員? Shouldn’t the person be the object?

With 聞く (to ask), the person you ask takes に, not を. Pattern:

Person に + thing (を / について) + 聞く Examples:
先生に道を聞く。 I ask the teacher for directions.
店員に無料修理について聞く。 I ask the clerk about free repairs. Using 店員を聞く is ungrammatical.

Does 聞く mean “to ask” or “to listen/hear”? How do I tell?

Both, depending on particles and context.

人に聞く = ask someone.
音楽を聞く = listen to music.
人から聞く = hear from someone. In your sentence, 店員に聞きました clearly means “asked the clerk.”

What changes if I use から instead of に with 店員?

店員に聞きました = I asked the clerk.
店員から聞きました = I heard it from the clerk. (The clerk told me.) So に focuses on you asking; から focuses on you receiving information.

Do I have to use について? What are alternatives?

Alternatives (nuance varies):

無料修理のことで店員に聞きました。 natural, conversational
無料修理に関して店員に聞きました。 formal/paperwork
無料修理の件で店員に聞きました。 businesslike Casually, if asking a specific question, you often embed the question:
無料修理があるか店員に聞きました。

Is ついて from a verb? Should I write a kanji for it?

Historically it comes from a verb (“concerning”), but today について is almost always written in kana. If you use kanji, the correct one for “regarding” is 就いて, not 付いて (which means “attached to”). Best practice: write it as について.

Should I say 店員さん instead of 店員 to be polite?

When referring to a clerk politely in conversation, 店員さん sounds softer than bare 店員. Other natural options:

店の人 (casual “store person”)
スタッフ (loanword “staff”) When calling a clerk, you usually say すみません, not “店員さん!”

How can I include the actual question I asked?

Embed the question with か or かどうか:

Yes/No: 無料修理が可能か店員に聞きました。 I asked whether free repairs are possible.
かどうか: 無料で修理してもらえるかどうか店員に聞きました。
Wh-question: 無料修理はいつまでか店員に聞きました。 (until when)

What’s the nuance difference among 無料修理, 無償修理, and 無料の修理?

無料修理: “free repair” (plain, common; signage/ads).
無償修理: “no-charge repair” (formal/legal/warranty contexts).
無料の修理: literally “repair that is free”; sounds a bit more explanatory in everyday speech. All are acceptable; choose based on formality.

Is this sentence natural as is?

Yes. It’s perfectly natural. Many people might also say 店員に無料修理について聞きました, which puts the person earlier, but your version is fine.

Why are there spaces between words here? Japanese usually doesn’t have spaces.

Right—standard Japanese writing does not use spaces. The spaces here are for learners to see word boundaries. In real text you’d write: 無料修理について店員に聞きました。

How do I read and pronounce the sentence?

Reading: むりょうしゅうりについててんいんにききました。
Romaji: Muryō shūri ni tsuite ten’in ni kikimashita. Note the pause in ten’in (n followed by a vowel).

How can I make this more polite/formal if I’m talking to the clerk?

When addressing them, use a humble verb or a polite request:

すみません、無料修理について伺ってもよろしいですか。
無料修理について教えていただけますか。 In narration (talking about what you did), 店員に聞きました is fine.

Can I use は to topicalize?

Yes, to emphasize or contrast:

無料修理については店員に聞きました。 (As for free repairs, I asked the clerk.)
私は無料修理について店員に聞きました。 (adds/clarifies the subject “I.”) Use は when you want a contrastive or topic focus.

What’s the difference between に and で here? Could I say 店で?

店員に marks the person you asked.
店で marks the place where you asked. You can combine them: 店で店員に無料修理について聞きました。 I asked a clerk at the store about free repairs.