It is permissible to look in the normal conventional way, but it must be without sexual enjoyment.

You can find out about how religious and chaste a girl is by asking her relatives and friends, and by asking about her family in general.

Yes, it is permissible to live like that as marriage is highly recommended and not compulsory, although Islam strongly recommends it in many narrations, like when the Messenger of Allah (peace be upon him) said: "There is no deed in Islam that Almighty Allah loves more than marriage."

The permission of the wife in the said case is required based on compulsory precaution.

Mutah is a marriage contract between a man and a woman where they agree on establishing a marital relationship with the stipulation of a dower, for a certain period of time. The said relation ends when the mentioned time elapses. Muslims have agreed on this legislation, but the second caliph prohibited it during his caliphate, and this became the reason for the differences in opinion on it, as history and jurisprudential books explained.

He is the father and the paternal grandfather.

Yes, it is permissible based on the said scenario.

Anything that has a value and is permissible to earn from, by selling or leasing for instance, can be set as dowry, even if it is a little. It can be a tangible item, or a debt that is becomes liable for, or a service, such as to sew a garment or recite the Quran.
As for those things which are not permissible to make one’s earning by, such as musical instruments or pigs, it is not permissible to make them the dowry.

The said marriage contract is valid.

The contract can be done by yourselves, but it is better to seek the help of a religious cleric.

The wife has the right to annul the marriage due to the insanity of the husband, whether he was insane before the marriage or became so after it.

Yes, he is allowed, whether his wife is away or not.

If her father and paternal grandfather have died, then there is no need to seek the permission of any of her relatives.

No, It is not required.

If she appointed him to act on her behalf and she approved the marriage, then the marriage is valid.

Muslims have unanimously agreed that temporary marriage was legislated by the Prophet (peace be upon him and his holy progeny). The second caliph stopped people from doing it during his rule at a famous incident. Since it was legislated by the Prophet, it remains lawful and valid. And this is what the followers of the Ahlul-bait (peace be upon him) and a group of the companions of the prophets (peace be upon him) believe.

It is permissible to marry an infertile woman, although it is better to choose a fertile woman as a wife.

The wife can ask the husband not to remarry again but it is not obligatory on the husband to accept her request.

If the mentioned person is ill in such a way that he loses his mind and he is considered to be insane, by acting in a way that sane people do not act, even for a limited period of time, then the wife has the right to annul the marriage. Otherwise, the marriage can only be ended through the husband divorcing the wife.
However, if the wife is willing to live with him and undertake her religious obligations to him and the husband is not willing to maintain her and provide for her, then the religious authority, the marja, can divorce the wife in this particular case if the husband refuses to divorce her.

It is permissible to marry a non-hijabi girl if she a Muslim. It is better to choose a believer that fulfils all her religious obligations as she will be the mother of your children in the future and children will be affect by the behavior of their mother.
At the meantime, you could increase your efforts in finding the suitable wife and maybe decrease other requirements in the wife will help you find one.