I’d rather have a room of my own, however small it is, than ___ a room with someone else.

选C
I'd rather do...than...是一个固定搭配，意思是“我宁愿做……也不做……”，这里中间有一个however small it is的插入语来修饰room.
句子翻译：我情愿有自己的房间——无论有多小——也不愿意与别人同房。

选择C
这是个短语 would rather do ...than do 宁愿....而不愿意....
中间however small it is不过是个插入语 无关紧要
我宁愿拥有一个我自己的房间，不管有多小，而却不愿意和别个合住。

would rather do something than do something

Example:
I would rather go home by bus than walk.

wounld rather = 'd rather

知识点：would rather do…… than do sth.宁愿……也不愿意……
我宁愿有个自己的小屋，无论多么小都没关系，也不愿意和别人分享一间屋子。

rather…than…词组前后所跟动词时态需一致。前面是动词原形have，故后面也要用动词原形share。翻译：我宁愿自己住一间房子，不管它有多小，而不愿与别人共处一室。

I would rather do than do sth.我宁愿做。。。也不愿做。。。（均用动词原形）