Prayer

It is permissible for the one who prayes tahajud to hold the Quran in any way it is suitable for him.

Yes, you have to read it all.

He must pray immediately after he woke up.

Praise be to Allaah. Shaykh Muhammad al-‘Uthaymeen was asked this question and said: The correct view is that this does not invalidate your prayer, because doubts often come to a man without him wanting them to. The Prophet (peace and blessings of Allaah be upon him) explained the ruling concerning doubts in prayer, and that […]

In sha Allah they will get the reward of it. You can search for it on the site.

This is not permissible. You have to pray in the masjid and go back home to give her a hand. It is only few minutes that you spend in praying.

This is permissible but not recommended as the rule is that the one memorizing more, leads the prayer. She is too young to pray and too young to demand such a thing. You should encourage her to learn more Quran and make that her incentive.

The prayer is valid unless you do something that nullifies it such as laughing out loud, talking, eating, etc. However, the lesser the Khushu the lesser the reward. https://www.youtube.com/watch?v=RtR_XuFz6WY

If one misses a pillar of his prayer such as one Rakah and doesn’t offer it shortly afterwards and does sajde sahoo, his prayer is invalid and he must pray it again.

This is not part of the authentic hadeeth! The last verse of the Surah pauses a question which is a statement actually. The question is: Isn’t Allah the best of judges? The weak narration says that worshippers answer this by saying: Yes, and I testify to that!