Women must cover their hair entirely during prayer; the prayer is valid if she noticed after the prayer that a bit of hair was not covered during the prayer.

Yes, this is fine.

During the prayer, it is obligatory for men to cover their private parts, which are the male reproductive organs and the anus. It is not obligatory to cover the rest of the body. The same is obligatory for men for outside the prayers.

Yes, it is permissible.

It is not obligatory for men to cover their heads during prayer.

The prayer is not invalidated by this, unless one is praying on the roof or the like, such that the private parts are considered exposed.

If you noticed during the prayer that the arm is visible, then based on compulsory precaution one should repeat the prayer. You can nullify the first prayer by doing what makes it invalid, like turning the body fully away from the direction of Qiblah, and then starting the new prayer.

Alcohol is Najis if it causes intoxication and the prayers is void when the clothes become Najis with alcohol or any other kind of Najasah. Yes, if the person does not know that alcohol is Najis, then the prayer is valid.

Yes, it is permissible to do so.

Yes, it is permissible and the prayer is valid in the said case.

No, it is permissible.

If the animal is slaughtered in a non-Muslim country then its leather is Najis unless the slaughterer was a Muslim in that country. Praying while wearing Najis things invalidates the prayer.

If the leather was from an animal that was not slaughtered in accordance with the Sharia, then wearing it during the prayer invalidates it.

Prayers when wearing such clothes are invalid.

Based on an obligatory precaution one should not carry a leather of any animal that was not slaughtered in accordance with the Sharia during the prayers.

If you offered your prayers with the leather belt out of ignorance then your prayers are valid. If you prayed out of forgetfulness of the law then they are invalid based on compulsory precaution.

The prayer is invalid if there was cat's hair on the person's clothes based on compulsory precaution.

Yes, the hair of a cat is regarded as Tahir unless it is known to have become Najis. But the prayer is invalid based on an obligatory precaution if the individual has on his body or clothes any part of an animal whose meat is impermissible to eat - like a cat - even if it is hair or fur.

Having nail polish is permissible during the prayer if it was applied after wudhu or ghusl. It prevents water from reaching the nail so it invalidates wudhu or ghusl if it was applied before performing any one of them. It can be applied on some of the toes that are not wiped during wudhu.

If the only clothing available is najis, offering the prayer in such clothing is valid in cases of cold weather or similar necessities. If it can be taken off, the prayer should be offered twice, once with it and once while naked, based on compulsory precaution.
The mentioned necessity rarely occurs in our time since making use of a covering material during prayer is easily done. If not usual clothing, it can be anything like sheets, blankets or similar things as long as the private parts (as defined by the Sharia) are covered for the males and all parts of the body that must be covered for females are covered.