Learn Hardcore Korean: gwangwanghamyeonseo hanguk eumsigeul manhi meogeosseoyo. - While sightseeing, I ate a lot of Korean food.

-면서: two actions happen simultaneously (관광하면서 먹었어요 = I ate while sightseeing).
-다가: one action was in progress and then something else happened or interrupted it. (영화를 보다가 전화가 왔어요 = I was watching a movie when the phone rang.)

관광하면서 = “while (I’m) sightseeing,” emphasizing simultaneous actions.
관광하고 = “sightseeing and …,” simply lists actions without implying they overlap in time.

Could we say 관광을 하면서 instead of 관광하면서?

Yes. 관광을 하면서 is more explicit (관광을 + 하면서) but means the same: “while doing sightseeing.” In casual speech Koreans often shorten it to 관광하면서.

Why is there no subject like 저 or 나는 in this sentence?

In Korean, subjects (저, 나는) are frequently omitted when the context makes it clear. Here it’s understood that “I” am the one sightseeing and eating.

What role does 많이 play before 먹었어요, and why is it placed there?

많이 is an adverb meaning “a lot.” In Korean, adverbs usually precede the verb they modify. So 많이 먹었어요 = “ate a lot.”