That's a very demanding question, but part of the issue with English history in particular is that Henry VIII separated the English Church from the Catholic Church (the one with the Pope) when he made himself its head in 1534. This meant that monarchs could rule the church as the pope did (James I is one who did). More on that in a moment.

1. Some kings had mistresses; some didn't. Henry VIII (1530's) and Charles II (1670's) did. James I (1610's) had male lovers. Charles I (1630's) is one king who seems to have respected the sanctity of marriage.

2. By queen, do you mean reigning monarchs (Mary I, Elizabeth I, Mary II, and Anne) or just the wives of the King (e.g. Anne of Cleves, Catherine of Braganza, and Mary of Modena)? In the former case, a woman sitting on the throne wielded a great deal of power, but still had to uphold a reputation for virtue; in society at this time it was more acceptable for a man to sleep around than for a woman to do so. This becomes clear in the latter case, if you imagine the political implications of any bastards she had, which would be indistinguishable from legitimate successors to the crown. This is why it was less common/okay for women to do.

3. Look up wikipedia's entry on Henry VIII's separation from the Roman Catholic Church, which kicks off a fascinating period in English history. The pope had granted annulments (a retroactive "undo-ing" of a Christian marriage) to various monarchs, usually because they wanted to remarry and ensure a succession, but he refused to grant one to Henry (being at that time imprisoned in the Vatican by Charles I of Spain and V of the Holy Roman Empire, Henry's wife's nephew). Therefore, Henry asked Parliament to declare him head of the Church of England, and dissolved his own marriage. After this point, there's really no appealing (at least in English politics) to an outside religious power. The grounds you describe, however, would not constitute legitimate grounds for an annulment, which requires some demonstrable evidence that marriage had never truly occurred, through something like deception, lack of either intention or ability to enter into marriage, or canonical impediment. Pragmatically, it wouldn't make sense for a queen to ask for a divorce either, since it would be seen as a selfish prioritization of her own ridiculous expectations over the international peace that her marriage had usually secured. When Charles II made his favorite mistress Barbara Palmer a Lady of the Bedchamber to his new wife Catherine of Braganza, relations between England and her home country of Portugal temporarily soured. A queen like Catherine learned to make the best of her situation, and could quietly avoid the king's company without much difficulty.

4. No; society has a high tolerance for the philandering of the powerful.

5. That's outside my pay grade, but I'm sure there's good literature on it and I'll poke around if I get the time later. It would help to know what period you're asking about.