Yes, "먹은" is the past-tense adjectival form of 먹다, as you say. -(으)ㄴ 후에 is the proper way to say "after (having done) something". In this case -ㄴ or -은 implies a completion (having done it) rather than the past (it can represent both the past and the present perfect). It might look odd because English doesn't have a similar structure, but this is the idiomatic way (think of it as "in the aftermath of having eaten the meal", if you can). For most words indicting "after", like 후, 뒤, 다음, 연후, etc. we use -(으)ㄴ, as it naturally expresses completion. We never say 밥을 먹기 후에 or 밥을 먹는 것 후에, even though they might look logical to a new learner. On the other hand, we never say -(으)ㄴ for anything indicating "before" since in this case the action hasn't even started, let alone completed. So we say -기 전에 or -기에 앞서서, as in 밥을 먹기 전에, 본론에 들어가기(에) 앞서(서), etc. If two things are happening at the same time, the proper verb ending is the "action" ending 는. so we say 먹는 동안에, 가는 중에/도중에, 하는 와중에, etc. You could also say -면서, as in 차를 타고 가면서. Nouns can be used with all the above expressions: - 식사 후에 = 밥을 먹은 후에. - 퇴근 뒤에 = 퇴근한 뒤에. - 식사 전에 = 밥을 먹기 전에 / 식사하기 전에. - 식사 중에 = 밥을 먹는 중/동안에 / 식사하는 중에.