Learn Hardcore Korean: yeogwon boksaboneun chekeu-in si yeonrakcheowa hamkke jechulhaeya haeyo. - You must submit a passport copy along with your contact information at check-in.

은/는 marks the topic, not the grammatical subject. Here, 여권 복사본 is the logical object of 제출하다, but it’s being topicalized: “As for the passport copy…”.
Using 을 is also fine and a bit more neutral: 여권 복사본을 체크인 시 연락처와 함께 제출해야 해요.
With 은, there can be a slight contrastive feel (e.g., “As for the passport copy, you must submit it…” possibly implying other items have different rules).

What exactly does 체크인 시 mean, and can I replace it with 체크인할 때?

체크인 시 literally means “at the time of check-in.” 시(時) is a bound noun meaning “time (when),” often used in formal/written language.
체크인할 때(에) means the same thing and sounds more neutral/natural in everyday speech.
Both are correct. On signs or notices, 체크인 시(에) is very common.

Should it be 체크인 시 or 체크인 시에? Is there any difference?

Both 체크인 시 and 체크인 시에 are acceptable. Adding 에 just makes the time marking explicit.
In formal writing, omitting 에 after 시 is very common (e.g., 예약 시, 결제 시).

Can I say 체크인에서 to mean “at check-in”?

No. 에서 marks a place, but here “check-in” is being treated as a time/event, not a physical location.
If you want a location, specify it: 체크인 카운터에서 (at the check-in counter).
For “when,” use 체크인 시 or 체크인할 때.

Is the 시 in 체크인 시 the same as the honorific suffix -시- or the clock 2시?

No. Here 시(時) is a bound noun meaning “time (when).”
It’s unrelated to the honorific -시- and only indirectly related to the “o’clock” 시 (both are written 時 in hanja but used differently).

Why is it 연락처와 함께? Could I just use 와 without 함께?

와 함께 means “together with” and forms an adverbial phrase: 연락처와 함께 제출하다 (“submit together with contact info”).
If you drop 함께, you must coordinate the objects: 여권 복사본과 연락처를 제출해야 해요 (“You must submit a passport copy and contact info.”).
여권 복사본을 연락처와 제출하다 is unnatural; use 와 함께 or coordinate both objects with 과/와 … 를.

What does 연락처 include—just a phone number?

연락처 means “contact information” in general. It often implies a phone number, but it can include email, messaging ID (e.g., Kakao), etc., depending on context.

Is 복사본 the same as 사본? Which is more natural with passports?

사본 = “copy” (general/official term). Very common in formal contexts.
복사본 = “photocopy” (emphasizes it’s a duplicated copy).
With passports, 여권 사본 is the most standard/idiomatic on forms and signs, but 여권 복사본 is also understood.

How does 제출해야 해요 express obligation? Are there alternatives?

-아/어야 하다 expresses obligation/necessity: 제출해야 하다 → polite 제출해야 해요.
More formal: 제출해야 합니다.
Very common variant: 제출해야 돼요 (same meaning in speech).
Request (softer): 제출해 주세요 / Polite formal request: 제출해 주시기 바랍니다 or 제출해 주십시오.
Honorific obligation (to customers): 제출하셔야 합니다 or 제출해 주셔야 합니다.

There’s no subject like “you.” Who is supposed to submit it?

Korean often omits obvious subjects. Here it’s an impersonal instruction; the understood subject is “you (the guest).”
On formal notices you might see an explicit, honorific subject: 고객님께서는 … 제출해 주셔야 합니다.

Can I drop some particles in this sentence?

In casual speech or on terse signs, particles are sometimes omitted, but clarity can suffer.
The given sentence is well-formed. If you simplify, prefer standard patterns like:
- 체크인 시 여권 사본과 연락처를 제출해 주세요.
- Avoid cramming nouns without needed particles.

How flexible is the word order here?

Elements before the verb are fairly movable. Natural variants include:
- 체크인 시 연락처와 함께 여권 사본을 제출해야 해요.
- 여권 사본은 체크인할 때 연락처와 함께 제출해야 해요.
Keep 연락처와 함께 before the verb it modifies. Don’t place it after the verb.
Using 은/는 tends to place that element earlier for topical emphasis.

Why is it 와 (not 과)? Could I use 하고/랑 instead?

과/와 are a formal pair: use 과 after a consonant, 와 after a vowel. 연락처 ends with a vowel sound, so 와.
In casual speech, 하고 or 랑 are common: 연락처랑 같이 or 연락처하고 함께.

Is 여권의 복사본 okay, or should it be 여권 복사본?

Both are grammatical. In noun-noun compounds, 의 is often omitted: 여권 복사본, 여권 사본.
여권의 복사본 can sound more formal/literary; in practice 여권 사본 is the most idiomatic for forms and notices.

How would this be phrased on a very formal hotel sign?

체크인 시 여권 사본과 연락처를 제출해 주시기 바랍니다.
Even more formal/firm: 체크인 시 여권 사본과 연락처 제출이 필요합니다. or 체크인 시 여권 사본과 연락처를 제출하셔야 합니다.