5. He did a research of the Darwin's theory of evolution.
他对达尔文的进化论做了研究。
还能这样说：
He engaged in researching in Darwin's theory of evolution.
He researched in Darwin's theory of evolution.
应用解析：
academic research 学术研究;
advanced research 远景研究，预测;
action research 动作研究;实践研究
6. I believe in Chinese Buddhism.
我信仰中国佛教。
还能这样说：
I have a strong belief in Chinese Buddhism.
I have a strong faith on Chinese Buddhism.
应用解析：
belief指承认某事是真的，尽管有或没有确凿的证据;
trust指信赖、信任，含有坚定的信念的意思;
confidence指在有证据的基础上相信，也常指自信、有把握。
7. We really have a good collection of cloisonne vases.
我们的确拥有种类繁多的景泰蓝花瓶。
还能这样说：
We have various kinds of cloisonne vases.
We have collected sorts of cloisonne vases.
应用解析：
make/take up a collection for 为……募捐
8. We owe much to Greek culture.
我们得益于希腊文化之处甚多。
还能这样说：
We benefit a lot from Greek culture.
We obtain a lot of benefits from Greek culture.
应用解析：
owe it to oneself 认为自己有必要，认为自己有责任;
owe sth. to 把……归功于，为……而应感谢