I come up with more than one words for each of those questions.
Plus, some of them don't sound quite right to me. You may want to consider changing your materials:
１）力がもう限界に（来て / 達して）いるので、とうとう倒れた。
(One was pushed to one's limits of one's strength, and one finally broke down.)
２）過去の教訓を真剣に（考える / 汲み取る）ベキである。
(One should seriously (consider / learn from) one's past lessons.)
--> One typo: ベキ (katakana) --> べき (hiragana).
３）若い頃、知識を（深める / 高める / 広める / つける / 吸収する / 豊かにする / 蓄える）のはとても大切である。
(It's very important for one to (deepen / heighten / broaden / acquire / absorb / enrich / accumulate) one's knowledge while one is young.)
--> 若い頃 sounds like an aged person looking back his/her youth ("when one was young"). I'd say 若いうちに instead ("while one is young").
４）謙遜に他人の作品のよいところを（認める / 取り入れる）。
(Humbly (recognize / learn from) the good points of other people's works.)
--> "謙遜に kenson ni" at the start doesn't sound quite right to me. It's usually used as a noun (謙遜 = modesty) or verb (謙遜する = be modest), but not as an adverb like this. We'd use "謙虚に kenkyo ni" to mean "humbly" instead.
５）バスがなかなかこなくて、気分が（むしゃくしゃして / ふさいで / 悪くなって）きた。
(The bus took forever to come (= was very late), and I started feeling (irritated / depressed / ill).)
--> There're very similar expressions:
- 気が滅入って ki ga meitte (feel depressed)
- 気が立って ki ga tatte (feel irritated)
but they take 気, rather than 気分, as the subject. Should the material say the correct answer is 滅入って or 立って, I have to say the material is not very reliable.