Learn Hardcore Korean: banggeum mun jamgasseo.

In casual speech, Koreans very often drop the object particle 을/를 when the meaning is clear. So 문 잠갔어 is perfectly natural. With the particle, 문을 잠갔어 sounds a bit more careful or emphatic on the door as the object. You can also use the topic marker for contrast: 문은 잠갔어 (As for the door, I locked it [but maybe something else I didn’t]).

What’s the subject here? How do I say “I” explicitly?

Korean usually omits obvious subjects. In everyday context, a bare past like 잠갔어 is often understood as “I locked (it).” If you want to be explicit:

Casual: 나 방금 문(을) 잠갔어.
Polite: 저 방금 문(을) 잠갔어요. Use 제가/나는/저는 if you want to emphasize or contrast the subject.

What’s the difference between 잠갔어, 잠겼어, and 잠겨 있어?

잠갔어: active/transitive “(I/Someone) locked (it).”
잠겼어: passive/inchoative “(It) got locked / became locked.” Often needs a subject marker: 문이 잠겼어.
잠겨 있어: state description “(It) is locked (right now).” Most natural for an ongoing state: 문이 잠겨 있어.

Is 잠궜어 a mistake? I hear it a lot.

Yes—avoid 잠궜어 in standard Korean. The correct past form is 잠갔어. You may hear 잠궈 (present connective) in speech, but the safe, standard forms are 잠가(요) and 잠갔어(요). In writing and careful speech, stick to 잠가 / 잠갔어.