Learn Hardcore Korean: gichayeok ape juchajangi isseoyo. - There is a parking lot in front of the train station.

에 is the locative particle. When attached to 앞 (front), it turns 앞 into 앞에, meaning “at/in front of” and marking that as the location where something exists.

Why do we use 에 instead of 에서 after 앞?

Use 에 for a static location (“there is”).
Use 에서 for the place where an action happens.
Since 있다 here simply states existence, 앞에 is correct.

Which word is the subject, and how is it marked?

The subject is 주차장 (parking lot).
Because it ends in a consonant, it takes the subject particle -이, giving 주차장이 to indicate “the parking lot” as the thing that exists.

What level of formality is 있어요, and why is it used here?

있어요 is the polite informal (존댓말) present-tense form of 있다.
It’s used in everyday conversation when you want to be polite but not overly formal.

How would you make this sentence more formal?

Replace 있어요 with the formal polite ending 있습니다:
기차역 앞에 주차장이 있습니다.

How would you make this sentence more casual/informal?

Drop the polite ending -요 (and optionally some particles):
기차역 앞에 주차장 있어.

How do you express the absence of the parking lot?

Use the negative existential 없어요 instead of 있어요:
기차역 앞에 주차장이 없어요.

Can you omit the locative particle 에 in colloquial speech?

Native speakers sometimes drop particles in very casual speech, e.g. 기차역 앞 주차장 있어.
However, for clarity and correctness—especially in writing or polite conversation—it’s best to keep 에.