Being under the water for 5 minutes would, in most, if not all cases make water reach all parts of your head, and that would be enough for your ghusl. The important thing is you follow the correct procedure of ghusl, which is head/neck, then right side of body, then left side of body.
Avoiding excessive doubt is also very important.
And Allah knows best
No, it is not obligatory to perform Salat before wearing Hirz of Imam Al-Jawad (AS).
It is not obligatory to take it off to go to shower.
When the shower water is continuous then it is connected with the Kurr and has the rule of Kurr water.
Yes. It is always very good to listen to Quranic recitation and remember Allah even while taking shower.
Ghusl does not need water flowing on the body, but needs just making any amount of water reach to every part of the body even by wiping water on all parts of the body. A big scholar in Najaf Ashraf who is Ayatullah Sayyed Muhammad Husain Al-Hakeem said: Ghusl can be performed with one cup of water by putting my hand in the cup of water then wiping over my head, face, neck, right side of body then left side of the body. Making the water touch by wiping to the whole body won't take more than one cup of water.
You can refer to Islamic rules of your Marja' of Taqleed for more details.
Ghusl does not need to be done in the shower, but it must be "washing", and water flowing, or to submerge yourself completely under water "ghusl irtimasi". Wiping in front of a sink would not be considered as ghusl, and therefore invalid.
And Allah knows best
It should be avoided as much as possible.
Merely touching one's private parts doesn't break Wudu for Shia Muslims; it would however break one's wudu for those who refer to themselves as Ahlus Sunnah. Yes, if one masturbated causing ejaculation this would break the wudu and one would require ghusl al janabah.
May Allah grant you success
Either is fine.
May you always be successful
No need to be dry before starting the Ghusl. We can perform Ghusl on wet body right after the shower.
This is one of differences between Ghusl and Wudhu. In Wudhu we must make the area of wiping (over the head and the feet) dry before wiping over it, but Ghusl does not need any part of our body to be dry.
After the Najasat is removed, shower water falling on the Najis areas is enough immediately to make your body Taahir with out need to go out of the shower and re enter.