1. 떡국을 먹고, 나이를 한 살 더 먹는다고 합니다.

=> They say we grow another year older when we eat rice cake soup (on the New Year's day).

=> They say we eat rice cake soup and grow another year older (on the New Year's day).

'-고' can be thought of as mentioning two loosely connected things one after another.

Your example is a subtle case because it is neither direct cause-and-effect nor unrelated events but something in between.

Also, -다고 합니다 means "..." 라고 사람들은 말합니다, or "They/People say ...", which should be included in the translation.

2. 하회 마을의 '하회'는 '물이 돈다'는 뜻으로 경치가 아름다운 곳입니다.

=> 하회 마을, where 하회 means "a river turns around", is a place of beautiful scenery.

=> 'Hahoe' meaning that "a river turns around", Hahoe Village is a place of scenic beauty.

하회 마을 is a name of a place, so it's optional to translate the 마을 part.

-(으)로 in '뜻으로' is a particle meaning "being ..." or "which is ...", a literary device like the participial clause or relative pronoun clause ", which is ..." in English.  It connects two factual clauses, the first one typically stating something introductory and the second one mentioning the same or related fact with greater detail.  It is similar to -(이)며 or  -(ㄴ/은/는)데 in a more colloquial style.

It is a common way to construct long, fact-heavy sentences.

For example, some 위키백과(Korean Wikipedia) pages start with such a sentence.

- 러시아 연방(...), 또는 러시아(...)는 동유럽과 북아시아에 걸쳐 있는 연방제 국가로, 세계에서 가장 영토가 넓다. 

- 그레이트브리튼 북아일랜드 연합왕국(...), 약칭 브리튼 또는 연합왕국(...) 혹은 영국(...)은 ... 주권국이자 섬나라로,  ...