Learn Hardcore Korean: jeoneun eoje deureun noraereul gieokhaeyo. - I remember the song I listened to yesterday.

In 어제 들은 노래, 들은 is the past attributive form of 듣다 (“to hear”). When you want to turn a verb into an adjective that describes a noun in the past, you attach -은 to the (possibly altered) stem. For 듣다, the ㄷ irregular changes to ㄹ, so the stem becomes 들. Adding -은 gives 들은, which literally means “(the song) that (I) heard.”

Why is 를 attached to 노래 instead of directly to 어제 or 들은?

The particle -를 is the object marker and goes on the head noun of the object phrase. Here, 어제 들은 노래 is one unit (a noun phrase modified by the relative clause). The entire phrase is the object of 기억해요, so -를 attaches to 노래, not to its modifiers 어제 or 들은.

What’s the difference between 저는 and 제가, and can I omit 저는 altogether?

저는 uses the topic marker -는, marking “I” as the topic (“as for me…”). 제가 uses the subject marker -가, focusing on “I” as the one doing the action. In everyday Korean, if it’s clear who you’re talking about, you can drop 저는 entirely:
어제 들은 노래를 기억해요.

What is the function of 어제 in this sentence, and could I move it elsewhere?

어제 is a time adverbial meaning “yesterday.” Inside the relative clause 어제 들은, it tells when the song was heard. If you move 어제 outside the clause, it no longer modifies 들은 directly. You could place it elsewhere as a sentence adverbial (e.g., 저는 어제 노래를 들었어요), but to keep “the song I heard yesterday,” 어제 must stay immediately before 들은.

Why is 기억해요 in the present tense? Doesn’t the sentence refer to something that happened yesterday?

The verb 기억하다 (“to remember”) describes a present state of mind. You say 기억해요 to mean “I remember (it now).” If you used the past 기억했어요, it would mean “I remembered (it) (at some point in the past),” which isn’t the same as currently recalling it.

Can I say 기억나요 instead of 기억해요? Are they interchangeable?

You can, but they’re different verbs. 기억나다 is intransitive—“to come to mind” or “to be recalled”—so you’d say:
어제 들은 노래가 기억나요.
기억하다 is transitive—“to remember something”—so:
어제 들은 노래를 기억해요.
Both convey that the song is in your memory, but the sentence structures differ.

What’s the difference between 어제 들은 노래 and 어제 들었던 노래? They both mean “the song I heard yesterday,” right?

Yes. 들은 = stem + -은; 들었던 = stem + -었-던. 들었던 can add a subtle nuance of an experience or emphasis on the past action, but in most contexts they’re interchangeable. Both mean “the song that I heard yesterday.”

Is word order flexible? Could I say 노래를 어제 들은 저는 기억해요?

No. In Korean, relative clauses must sit immediately before the noun they modify. “어제 들은” must directly precede “노래”. You can move 저는 or other adverbials around (저는 어제 들은 노래를 기억해요), but 어제 들은 노래 must stay together.