"갑자기 큰 폭풍이 몰아쳐 배가 부서져 버렸어요." => All of sudden, a big gale came upon us and the boat shattered.
Q1. 부서지다 and 버리다 are not the same. 부서지다 means "shatter; break apart", and 버리다 has two major uses: 1) independent verb meaning "throw out", and 2) auxiliary verb combining with another verb to add the sense of "done irreparably", "for good", or "completely", similar to English's "out" ("wipe" vs "wipe out" for example).
- 난 시험을 망쳤다 (I messed up the exam) -> 난 시험을 망쳐 버렸다 (I messed up the exam completely - more natural).
- 배가 부서졌어요 (The boat shattered) -> 배가 부서져 버렸어요. (It was shattered completely).
Q2. Yes, verbs that attach to another to modify its meaning are called 보조 동사(auxiliary verb). These are widely used for a lot of situations. Common examples include: 가다, 보다, 하다, 버리다, 싶다, 지다, 싫다, 않다 among many others.
- 책을 다 읽어 간다: I am almost finished reading the book (가다: "continue to; get close to").
- 한번 해 봐: Give it a try. (보다: "do for a trial; try out").
- 얼굴이 곱지 않다: The face is not fair (않다: "not").
- 뭔가 먹고 싶다: I crave something to eat (싶다: "want to", "desire").
- 아무것도 먹기 싶다: I don't feel like eating anything (싫다: "don't want to")
Q3. Multiple verbs may be used in the same phrase, but they are connected through connectives (i.e, verb conjugative endings playing a conjunctive role). Also, main + auxiliary forms like ~어 버리다 may be treated as one verb as they form a single unit of meaning.
The two main verbs 몰아쳐 and 부서져 버렸어요 are connected with the ~(아/어) connective in 몰아쳐:
- ... 몰아쳐 배가 부서져 버렸어요 == 몰아치(다) + ~(아/어) + 배가 ... (~(아/어) connective)
* ~(아/어) is similar to "and", a loose cause and effect connection.
Using such connectives, many verbs can be used in the same sentence:
- 나는 밥을 먹고 낮잠을 좀 자고는 운동하러 밖으로 나갔다 => I ate, took a nap for a bit, then went out to work out.
Using multiple verbs mostly involve the connectives / conjugations (not adjectives or 하다 you mention).
So, knowing these ~(아/어), ~고, ~지, etc forms are the key to mastering them.