完成进行式？

不行，如果选B，那就是前后两个句子了，两个句子之间用逗号连接，没有联接词，是不行的。
此空，应该填非谓语动词，是个分句，主语是the woman,是主动态，选d

选d ，逗号前才能整体作为状语。

B当然不可以
这样的话这个句子就有两个谓语了
选D

这个句子的完整说法是:
After she had finished all the housework,the woman turned on the TV and took a short rest.

选B的话,首先时态不对,应该用过去完成时(过去的过去),另外句子成分也不完整--没有连词,没有主语.

D中用having done,是一种分词短语代替状语从句的情况,是正确的.

类似的还有:

When she was walking in the park,she met an old friend.
Walking in the park,she met an old friend.(表示时间)
她在公园散步的时候，遇见一位老朋友。

As I have lived in the city for many years,I knew it well.
Having lived in the city for many years,I knew it well. (表示原因)
我已在这座城市居住多年，对它非常熟悉。

If you go there by plane,you will arrive tomorrow morning.
Going there by plane,you will arrive tomorrow morning. (表示条件)
如果你乘飞机去那里，明天上午就能到达。