Question

This cat is said to have been given to her不定式做什么语

Answer 1

This cat is said to have been given to her
这里 to have been given her 作句子的保留宾语。
因为这句是被动语态的语法结构。所以有必要把它逆推为正序形式。
正序的主动语态是：Someone said the cat has been given to her.
当主动语态变为被动语态时，正序语态句子的宾补要给予保留，但前面要加to, 由于
to 后面必须跟动词原形，（尽管这里的to是介词），所以是have been given to her.

Answer 2

亲爱的楼主，这个句子是错误的。say是不及物动词，不能跟名词或代词作宾语，只能跟句子作宾语，以上楼主所言把this cat当成了主动结构的宾语，显然是错误的。要想表达一个被动的说法应该：It is said
this cat has been given to her.据说这只猫已经送给了她了。

Answer 3

This cat is said to have been given to her
仍然是主语，因为这个句子是经过移位的，原来结构是；
it is said that this cat was given to her，it为形式主语，hat this cat was given to her为主语从句。

Answer 4

不定式短语 to have been....her 在句中充当宾语。
主语是 This cat
谓语是 is said

Answer 5

不定式短语 to have been....her 在句中充当状语