1. “In answering this question, I would like to cite here one of my earlier answers:

In the classical manuals of Fiqh wiping is generally taken to be restricted to leather socks/stockings. However, there is nothing in the sources to restrict the permission to leather only. The original intent of the permission was to bring ease and comfort; people normally wear socks/stockings regardless of the material they are made of in order to ward off dust, and protect the feet from heat or cold. Since it is inconvenient to take them off all the time, permission was granted by the Prophet (peace be upon him) to wipe over them for a period of one day and night for a resident and three days and nights for a traveler.

That the permission to wipe on socks/stockings is not limited to leather is the view authenticated by great scholars and Mujtahids such as Ibn Taymiyyah and others; the reasoning is quite clear: The Prophet (peace be upon him) never stated that the permission was limited to leather stockings. Therefore the same rule can be extended to all types of socks/stockings regardless of whether they are made of leather, cotton or wool, so long as it is Halal (permitted) material. There are, however, certain conditions to be considered for the permissibility of wiping such as the following: a); socks/stockings must be thick and not transparent so that water does not seep through when wiped over them; b); they must cover both the upper and the lower parts of the feet up-to the ankle-bones; c); they must be worn after purification; d); one must not take them off; e); wiping must not exceed the period of one full day and night in the case of a resident and three days with nights for a traveler. Finally, one must remove them if Ghusl (bathing) becomes obligatory.

To conclude: It is permissible to wipe on thick cotton or nylon socks/stocking. In case of those who are working in offices or factories or campuses, it may even be better for them to make use of this allowance in Shari`ah, especially where washing feet may create undue hardships or becomes a source of undue misgivings, etc. It is best for them to put on their socks after having completed ablutions in the morning; then they can continue wiping on them during the duration of their work or absence from home for a whole day or until they return home.”

2. I do not quite understand this question: If my understanding is correct, you should correct the mistake as soon as you are aware of it in prayer; if the imam has made a mistake, the follower has to correct him. If he became aware of the mistake later, then he should offer two prostrations before salam; this is known as sujud al-sahw.
3. If water is not available then one should wipe the penis clean with toilet paper and perform tayammum. When water becomes available, one should wash the genitals with water and perform wudu.