Learn Hardcore Korean: sangsaga bonaen imeireul ilgeoyo. - I read the email that the boss sent.

읽어요 is the polite informal (해요) form that you’d use in everyday conversation.
읽습니다 is the polite formal form, often used in announcements or presentations.
읽는다 is the plain (prosaic) form, typically found in written narration or diaries.

How does 상사가 보낸 이메일 translate to “the email that the boss sent”?

This is a Korean relative clause. The structure is:
(subject + subject marker) + (verb in attributive form) + (noun).

상사가 marks “boss” as the subject of the clause.
보낸 is the past attributive form of 보내다 (“to send”), modifying the noun.
이메일 is the noun being modified.
Put together, 상사가 보낸 이메일 literally means “boss-ga sent email,” i.e. “the email that the boss sent.”

Why is 보낸 used here instead of 보내는 or 보냈다?

보낸 is the past attributive form (action completed) used to modify a noun.
보내는 is the present attributive form, meaning “the email that the boss sends” (habitually or right now).
보냈다 is the past declarative ending, i.e. “(he) sent (it),” but it doesn’t modify a noun directly.

What role does the particle 가 play in 상사가? What changes if I use 상사는 instead?

가 is the subject marker, indicating that 상사 is the subject of 보낸. It simply introduces new information.
는 is the topic marker. 상사는 보낸 이메일을 읽어요 would shift emphasis to “As for the boss, I read the email they sent,” implying contrast or topic focus.

Why do we attach 을 to 이메일 in 이메일을 읽어요?

을 is the object particle. It marks 이메일 as the direct object of 읽어요 (“to read”). Without it, the sentence would be grammatically incomplete or ambiguous.

Why is 저 or 제가 omitted in 상사가 보낸 이메일을 읽어요? Do I need to say “I” explicitly?

Korean often drops the subject when it’s understood from context. Here, it’s clear the speaker is the one reading, so 저/제가 isn’t necessary. You can add 제가 for emphasis—제가 상사가 보낸 이메일을 읽어요—but it sounds redundant unless you’re contrasting with someone else.

How would I say “I am currently reading the email that my boss just sent”?

Add a time adverb and adjust the clause for “just sent”:

방금 (just now) or 지금 (right now) for timing.
Stick 방금 before the clause and 지금 before the verb:
“방금 상사가 보낸 이메일을 지금 읽어요.”
= “I’m reading right now the email my boss just sent.”