Should monks be envolved with Politics or Governments? I think the answer is a clear NO, but I have seen things like that so I would like to doublecheck and ask where is this "rule"? Vinaya?
In DN2, the Samaññaphala Sutta: The Fruits of the Contemplative Life, the Buddha speaks of virtues in three sections of his discourse; The Lesser Section on Virtue, the Intermediate Section on Virtue, and The Great Section on Virtue. Part of the Intermediate Section on Virtue reads:
"Whereas some brahmans and contemplatives, living off food given in faith, are addicted to talking about lowly topics such as these — talking about kings, robbers, ministers of state; armies, alarms, and battles; food and drink; clothing, furniture, garlands, and scents; relatives; vehicles; villages, towns, cities, the countryside; women and heroes; the gossip of the street and the well; tales of the dead; tales of diversity [philosophical discussions of the past and future], the creation of the world and of the sea, and talk of whether things exist or not — he abstains from talking about lowly topics such as these. This, too, is part of his virtue.
If even speaking of kings, ministers of state, armies, and battles is considered lowly, then it seems implicit that to be involved in politics and governments would also be lowly and not virtuous for one "living off food given in faith" i.e. an almsman or monk.