1. I was punished at school. - it is Past Simple Passive (you are right). according to Russian this sentence can be translated in two ways: 1. Я был наказан в школе. 2. Меня наказали в школе. (the second variant of translation sounds more natural, but both are correct and are in use)
2. I punished at school. - yes, you are right - if you write the sentence this way, it will mean that it were YOU who had punished someone at school.
3. to be allowed to - it is a passive construction...the synonym of the modal verb MAY - as MAY does not have a future tense (we can not say "will may") - "to be able to" can be used for this purpose. (E.g. I may open the window. I will be able to open the window)
4. Do you think teachers should be allowed to punish school children? - Вы думаете/считаете (suppose) учителям следует разрешить/позволить (to let) наказывать школьников?
5. The police take your fingerprints if you are arrested. - Полиция берёт ваши/у вас отпечатки пальцев, если вы арестованы/если вас арестовали. - both variants are natural for Russian. (it is Present Simple Passive)
*если вас арестовывают - will be "if you are being arrested"