Learn Hardcore Korean: more hangugeo siheomi isseoyo. - There’s a Korean test the day after tomorrow.

Korean often uses a time expression instead of a verb tense to show future events. By placing 모레 at the beginning, you mark the time (“the day after tomorrow”), so 있어요 stays in the present form, which literally translates as “(An exam) exists the day after tomorrow,” i.e. “I have an exam the day after tomorrow.”

What role does 이 play in 시험이 있어요?

이 is the subject particle attached to nouns ending in a consonant (like 시험). It marks 시험 (“exam”) as the subject of 있어요 (“to exist”/“to have”).

Why is 있어요 used here, and what other forms could appear?

있어요 is the polite informal form of 있다 (“to have” or “to exist”).
In a formal setting you’d use 있습니다.
In casual speech you might say 있어 or even just 있어? for “Is there…?”

Could I reorder the sentence, for example say 한국어 시험이 모레 있어요? Does word order matter?

Korean uses particles (이/가, 은/는, etc.) so word order is relatively flexible.
All of these are grammatically fine and mean the same:
• 모레 한국어 시험이 있어요.
• 한국어 시험이 모레 있어요.
• 시험이 모레 한국어 있어요. (the last one is awkward—time + topic/object should come before the marked subject)
Putting 모레 first is most natural for emphasizing the time.

Why is it 한국어 시험 and not 한국말 시험?

한국어 is the standard term for “Korean language,” especially in formal or educational contexts.
한국말 is more colloquial, like “Korean speech.”
For an exam context, 한국어 시험 sounds more natural.

Why use 시험이 있어요 instead of the verb 보다, as in 시험을 봐요?

시험을 보다 literally means “to take an exam.” It focuses on the action of taking.
시험이 있다 literally means “an exam exists/is scheduled.” It focuses on the existence of the exam.
When announcing or asking about scheduled exams, Koreans often say 시험이 있어요 (“There is an exam…”). When you talk about actually taking it, you say 시험을 봐요 (“I take/do the exam”).