What is the ruling in case of a woman who due to cancer was unconscious? Does she have to give compensation for the missed Salah? What about Ramadan fasts?

What type of compensation are you talking about? If a person misses Salah due to unconsciousness then those Salawaat are waived, if omitted for a reason but was not aware of it, such as unconsciousness. In this case the prayer is waived for her and she does not have to make it up.

The scholars of the Standing Committee were asked:

 ‘I was involved in a car accident and I stayed in the hospital for three months, where I did not wake up and I did not pray for all that time. Is it waived for me or do I have to make up all the missed prayers?’

They replied: ‘The prayers during the period mentioned are waived in your case, so long as you were unconscious throughout that period.’

They were also asked:

‘If a person remains unconscious for a month and does not pray throughout that period, then he regains consciousness after that, does he have to repeat the missed prayers?’ 

They replied:

‘He does not have to make up the prayers that he missed during that period, because he comes under the same ruling as one who is insane, and the Pen is lifted from one who is insane (i.e., his deeds are not recorded).’

Source:

[Fataawa al-Lajnah al-Daa’imah, 6/21]