Learn Hardcore Korean: hyudaepone bimilbeonhoga seoljeongdwae isseoyo.

It marks the location where a state exists. With 있다, 에 often translates as on/at/in depending on context. So 휴대폰에 … 있어요 means the state exists on the phone. Use 에, not 에서, because this is a state, not an action in progress.

Why is 비밀번호 marked with 가 instead of 을/를 or 은/는?

가 marks the grammatical subject: 비밀번호가 (subject) 설정돼 있어요 (is set).
을/를 would be for the object of an active verb (e.g., 비밀번호를 설정했어요 I set a password).
은/는 topicalizes: 비밀번호는 설정돼 있어요 = As for the password, it’s set (contrastive).

What exactly does the pattern -어/아 있다 mean in 설정돼 있어요?

It expresses a resulting state from a completed action. Compare:

설정됐어요 = It was set (event/completion).
설정돼 있어요 = It is set (resulting state continues). Don’t use -고 있어요 here; 설정하고 있어요 would mean someone is in the middle of setting it.

Why passive 설정되다 instead of active 설정하다?

Because the sentence describes the phone’s state, not who did the action. 비밀번호가 설정돼 있어요 = The password is set. If you want to say who set it, use the active: 제가 비밀번호를 설정했어요 (I set the password).

How would I say “I set a password on the phone”?

Neutral: 휴대폰에 비밀번호를 설정했어요.
With the nuance “I set it and left it that way on purpose”: … 설정해 놨어요 or colloquially 비밀번호를 걸어 놨어요.

Is 설정돼 있어요 the same as 설정되어 있어요?

Yes. 돼 is the contracted form of 되어. 설정돼 있어요 is very common in speech; 설정되어 있어요 can feel a bit more careful/formal, but both are correct.

How do I pronounce and spell 돼 correctly?

It comes from 되 + 어 → 돼 and is pronounced roughly “dweh.”
Write 돼요, not the common misspelling 되요.
돼 있어요 is pronounced like “dweh i-sseo-yo.”

Can I say 휴대폰에 비밀번호가 설정돼요?

Not for this meaning. 설정돼요 sounds like “it gets set (habitually/regularly).” To describe the current state, use 설정돼 있어요.

How do I make this more casual or more formal?

Casual: 휴대폰에 비밀번호가 설정돼 있어.
Polite: … 있어요. (original)
Formal: 휴대폰에 비밀번호가 설정돼 있습니다. or 설정되어 있습니다.

How do I say there’s no password set?

Standard: 비밀번호가 설정돼 있지 않아요.
Colloquial: 비번이 안 걸려 있어요.
In everyday speech you’ll also hear: 비밀번호가 없어요 (There’s no password), though it’s less technical.

Is the word order fixed? Can I move things around?

You can move the location phrase:

비밀번호가 휴대폰에 설정돼 있어요 (also natural). Adding topic for contrast is common:
휴대폰에는 비밀번호가 설정돼 있어요, 태블릿에는 없어요.

Could I use 있다 without 설정되다, like 휴대폰에 비밀번호가 있어요?

You can, and it means “There is a password on the phone,” but it’s vaguer. 설정돼 있어요 clearly states the setting is configured.

What’s the difference between 설정돼 있다, 잠겨 있다, and 걸려 있다?

설정돼 있다: the setting is configured (neutral/technical).
잠겨 있다: it is locked (focus on lock state; doesn’t specify it’s by password).
(비밀번호가) 걸려 있다: colloquial “a password lock is on.” You’ll also hear 잠금이 걸려 있어요.

Why not 계시다 instead of 있다 for politeness?

계시다 is the honorific form of 있다 for animate subjects (people). Here the subject is 비밀번호 (inanimate), so use 있다, not 계시다.

Why use 에 and not 에서 here?

에 marks the location of existence or a state. 에서 marks where an action occurs. Since this is a state (… 있어요), 에 is correct.

What are the nuances of the phone and password words here?

휴대폰 ≈ mobile/cell phone; 휴대전화 is a slightly more formal synonym; 스마트폰 specifies a smartphone.
비밀번호 (standard), 비번 (slang/short), 암호 (code/password, slightly more formal/technical), 패스워드 (loanword). All can fit, but 비밀번호 is the most common on devices.

Can I add a possessor like “my”?

Yes:

Polite: 제 휴대폰에 비밀번호가 설정돼 있어요.
Casual: 내 휴대폰에 비밀번호가 설정돼 있어.