Learn Hardcore Korean: eoneu sikdangeseo babeul meogeoyo?

어느 is an attributive pronoun meaning which and always modifies a noun—in this case 식당. You use it when you want to choose one item out of a known set. 어디, on the other hand, means where and stands on its own without modifying a noun.
Examples:

어느 식당 = which restaurant
어디에 가요? = Where are you going?

Why is the particle 에서 used after 식당, and not 에?

The particle 에서 marks the location of an action. Since 먹어요 is an action verb (to eat), you need 에서 to indicate where that action takes place:

식당에서 밥을 먹어요 = eat at a restaurant
In contrast, 에 marks location of existence or direction:
집에 있어요 = am at home
학교에 가요 = go to school

What does the particle 을 do in 밥을 먹어요, and how is Korean word order different from English?

The particle 을 marks 밥 as the direct object—it answers what you eat.
Korean follows Subject-Object-Verb (S-O-V) order, and you often place locative or adverbial phrases before the object:
(Subject) + 식당에서 (location) + 밥을 (object) + 먹어요 (verb)
In English, you use Subject-Verb-Object and often put the location phrase at the end: “I eat rice at the restaurant.”

Why is the word 밥 used here to mean meal instead of specifically rice?

In Korean, 밥 literally means cooked rice, but it is commonly used to refer to a meal in general. Saying 밥을 먹다 is the standard expression for to have a meal, regardless of the food.

Why is there no subject in 어느 식당에서 밥을 먹어요?

Korean frequently omits the subject when it’s understood from context. If you’re asking a friend about themselves, the subject is implicitly you; if you’re talking about yourself, it would be I. You omit it because the listener already knows who is performing the action.

What level of politeness does 먹어요 convey? Could I use another ending?

먹어요 is polite informal speech (해요체), suitable for everyday conversation with strangers, colleagues, or people of similar status. Other options include:

먹습니다 (formal polite, 합쇼체)
먹어 (casual, 반말)
먹습니까? (formal question)

Is it possible to use a different verb like 식사하다 instead of 밥을 먹어요?

Yes. 식사하다 is a more formal or neutral verb meaning to dine or to have a meal.

어느 식당에서 식사해요? (polite informal)
식사해요 is still polite informal like 먹어요, but slightly more elevated in style.

How might I answer 어느 식당에서 밥을 먹어요? in Korean?

You can name the restaurant, then add 에서 먹어요. For example:

저는 김밥천국에서 먹어요. (I eat at Kimbap Heaven.)
우리 집 근처 식당에서 먹어요. (I eat at the restaurant near my house.)