1- Madhi is considered filthy not clean as most scholars rule. As also in the hadith by Imam Ali (may Allah be pleased with him) sent another Sahabi by the name of Al-Miqdad (may Allah be pleased with him) to ask the prophet (peace be upon him) about madhi and the Prophet (peace be upon him) said: “let him wash his penis and make Wudu.” (Al-Bukhari)

 

Based on this, you can go to the washroom in the school, wash your private area, and since you might not have an extra underwear, you will need to wash the area of the underwear itself touched by madhi and then make wudu’ and you can perform your prayer insha Allah.

 

2- Washing machines are better more cleaning methods than formal hand washing old fashion way. It does consist of water, detergent solution and long periods of cleaning movement cycles and rinsing etc. So yes it is sufficient.

 

3- Wiping over socks is something the Prophet (peace be upon him) did and scholars approved in consensus but with some details in conditions.

 

The conditions are:

 

1- You must wear them after you are on wudu’.

 

2- Socks must cover all feet till over ankles. Avoid those short socks under ankle.

 

3- They should not be made of pure silk.

 

4- That they are thick and that you can walk with them (some scholars said it’s not a condition).

 

5- Finally, if you are travelling, you can do it for three days, if you are not travelling just for one day.