Islamic education for one's child

Q: A father said that he will admit their child in a madrasah, but the mother said that she will not admit their child in a madrasah. The mother respects Allah and Islam, she also wants the child to know about Islam and she wants to teach the child about Islam by an Islamic teacher. The father and mother then admitted their child in a school (not an Islamic school and not a madrasah) and appointed an Islamic teacher to teach their child how to pray salaah, read Quran, duas, Islamic characters, lifestyle, etc. so that child learns about islam. Is there any sin?

Experiencing a slight discharge after thinking about something

Q: I would like to know if ghusl is fardh on a woman if she has a thought and feels a slight discharge, but doesn't check it because on every ocassion it is seen to be normal discharge, and not mani.

I always experience this during istinja and have checked on many ocassions, but have recently stopped checking because I experience a lot of normal discharge and know it can't be mani. I am still doubtful. Am I ritually impure for not checking this time?

Big bang theory

Q: I have a few questions based on the scientific theory of the Big Bang. I have heard some people using the following 2 aayaat to prove the scientific theory of the Big Bang

(١) أَوَلَمْ يَرَ الَّذِينَ كَفَرُوا أَنَّ السَّمَاوَاتِ وَالْأَرْضَ كَانَتَا رَتْقًا فَفَتَقْنَاهُمَا ۖ وَجَعَلْنَا مِنَ الْمَاءِ كُلَّ شَيْءٍ حَيٍّ ۖ أَفَلَا يُؤْمِنُونَ.

(٢) ثُمَّ اسْتَوَىٰ إِلَى السَّمَاءِ وَهِيَ دُخَانٌ فَقَالَ لَهَا وَلِلْأَرْضِ ائْتِيَا طَوْعًا أَوْ كَرْهًا قَالَتَا أَتَيْنَا طَائِعِينَ.

Based on this I have the following questions

1. What is the Islaamic perspective with regards to the manner the earth was created ?

2. Is there any way to prove the Big Bang theory using the aforementioned aayaat?

3. If a person openly proves and advocates the Big Bang theory using the aforementioned aayaat, but believes that Allah Ta'ala was the being who caused the Big Bang. What is the state of the imaan of such a person?

4. If a person believes that the Big Bang really occurred and that it was purely a matter of nature's occurrence and not at all the doing of Allah Ta'ala, then what is the state of the imaan of such a person?

Delaying in shaving the head after umrah

Q: My friend came out of ihram without shaving his head. He was willing to shave his head but someone misled him that it's not necessary if umrah was done once before. He completed his sa'ee and asked someone to shave his head but he didn't understand or whatever that person said. He came to the hotel room around 2 am and in the morning I asked him why he didn't shave his head so he told me that I asked the guard in Makkah where to cut my hair and they said no need.

Anyhow in the morning I told him that its necessary to cut/shave the hairs of your head and around 10 am or 11 am he shaved his head. He was out of ihram for about 9 hours. What is the penalty, dam or something else?