Learn Hardcore Korean: siheom junggane hyusik siganeul jwoseo joha. - I like that they give a break in the middle of the exam.

휴식 시간 is a noun phrase meaning “break time.” You need 시간 after 휴식 to specify a duration (“time”).
쉬는 시간 is also possible (“time for resting”), but 휴식 시간 sounds slightly more formal.
You cannot drop 시간 if you want to emphasize the period given; 휴식을 주다 without 시간 is incomplete.

What is the function of -줘서 in 줘서 좋아?

-줘서 is the contracted colloquial form of -주어서, which is the connective form of 주다 (“to give”) plus -서 (“because/and”). It links the giving of break time to the resulting feeling.
You could say:

휴식 시간을 주어서 좋아. (more formal)
휴식 시간을 줘서 좋아. (colloquial)

Could you use -주니까 instead of -줘서?

Yes, but nuance changes:

줘서 좋아 = “It’s good because they give a break.” (focus on cause + result)
주니까 좋아 = “When they give a break, I like it.” (focus on conditional “when/if”)
Both are correct, but -서 is more natural for simple cause–effect statements.

Why is 좋아 used here? Is it an adjective or a verb?

In Korean, descriptive verbs (adjectives) like 좋다 conjugate like verbs. Here, 좋아 is the present-form of 좋다, meaning “is good” or “I like it.”

There is no subject in this sentence. Is that okay?

Yes. Korean often omits the subject if it’s clear from context. Here, it’s implied that (시험 감독관/교사들이) give the break. If you want to be explicit, you could say:
시험 감독관이 시험 중간에 휴식 시간을 줘서 좋아.

Can you omit the object marker -를 in 휴식 시간을 줘서?

In standard Korean, you normally keep -를 to mark the object (휴식 시간을). In very casual speech, people sometimes drop particles, but for clarity and correctness it’s best to include -를.

Is there a nuance difference between 시험 중간에 and 시험이 중간에?

시험 중간에: “in the middle of the exam.” Smooth and natural.
시험이 중간에: awkward, because 시험이 would mark “exam” as the subject, and you’d need a verb or adjective (e.g., 시험이 중간에 어렵다). To indicate time or point, stick with 시험 중간에.