你的分析大部分是正确的，个别地方有问题，我改正如下：

1. He likes to half close his eyes.

主语：He

谓语：likes

宾语：to half close 改为：to half close his eyes

状语：half 是用来修饰动词close

定语：his eyes 改为：his eyes 作close的宾语

2. Our object is to further cement friendly relations between the two countries.

主语：our object

谓语：is to further cement, 这里的is不是主系表结构中的系动词，应该是 be to do表将来时态，改为：is 至句末为“系表结构”。is 是系动词；to further cement friendly relations between the two countries 是表语，其中further作状语修饰cement，friendly relations between the two countries为cement的宾语，其中，friendly和between the two countries分别作定语修饰relations

状语：friendly（删去）

relations between the two countries 我不晓得这里是做啥句子成分了！
3.  At first he wasn't interested in bookkeeping, but later he began to actually like it.

时间状语：at first

主语：he

谓语：动词短语 wasn't interest in doing sth. 改为：wasn't interested in bookkeeping 是谓语部分，也叫“系表结构”。

宾语：bookkeeping

时间状语：but later 改为：but 是并列连词引导第2个分句，表示转折，later 是时间状语。

主语：he

谓语：began

状语：to actually like it 改为：作began的宾语。