If you are Baaligh, you must then fast the whole month of Ramadan and not only twenty days. Your parents have no right to stop you from obligatory fast.
If you are still not Baaligh, then you should listen to your parents.
Yes of course. Acceptance of every worship depends on the sincerity in the intention.
Menstruating female is allowed to perform all worship e.g. Thikr of Allah, Du'a, reading Quran (except the four verses of obligatory Sojood). She is not allowed to pray any Salah nor perform Tawaf around Ka'ba nor stay in Masjid.
Illness has it's rules in Islam, including not fasting if illness can be initiated or worsened by fasting. I'll person who can not fast is not going against the orders of Allah, but obeying Him by not fasting.
Among the necessary things to observe between opposite genders is to avoid what we call khalwah. Khalwah means to be isolated and in a private closed off area with a non-mahram person. This is haram. When you are secluded with a non-mahram woman, within a particular setting, like what you mentioned, then there will be strong potential to engage in unlawful actions.
The hadith says that whenever there are two individuals of opposite gender alone, the third will be shaitan.
This is assuming that it's only you and her in the house. If there are other people, then it would not be considered as khalwah.
This is also not applicable to professional communication, like a work colleague in a meeting, or a counseling session, etc.
As Muslims, we must be extra cautious and very strict in our interactions with non-mahrams.
That being said, your fasting during the holy month of Ramadhan will be valid, as long as you observe all required conditions for fasting.
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