I suppose there are 14 questions in there. I'm basically looking for the events in each country that led to powerless monarchs.
Are there any grand lessons to learn about how monarchs lose power?
The very, very short answer to why they're still around is because no one saw fit to overthrow them. The English monarch lost power gradually but England still has a significant aristocratic faction. In the odd man out case, the Spanish monarch was reinstated in a fit of historical pride following fascism. On the other hand, the French removed their king (although briefly had an emperor) and the Russians removed their tsar, alongside several other kings of the east. Compare to Germany, who had an emperor at one point, but the position was dissolved (then reinstated and then finally dissolved).
Why? Because of socioeconomic forces favoring democracy, and violence. Monarchies which were not deposed have tended to stick around as cultural institutions, even through the European movement for democracy. Why such a movement? That's a big enough question for its own book, let alone thread.
The Russian institution is the one I'm most familiar with, but I know enough about the Spanish one, and it's an interesting case. The Second Spanish Republic was not a period of particular stability, but in theory it saw the throne's abdication. Franco later overthrew the Republic, such as it was, and actually styled himself as Regent. His successor Juan Carlos I, chosen by Franco, styled himself a king and drafted their current constitution. He was so wildly popular and respected for his actions that the monarchy was legally granted some powers in that constitution.
The Belgian monarchy was established with the Belgian state in 1830 (the first king was sworn in in 1831) as a constitutional monarchy. It was never an absolute monarchy. The role of the king was always largely ceremonial and circumscribed by the constitution.
hi! the FAQ (link on sidebar) has several threads on question 1 for the UK
The British got rid of their monarch (Charles I) by executing him. We then found out that a dictator (Oliver Cromwell) was worse. So we installed Charles II as a monarch with limited powers.
For the Netherlands it is very easy to answer.
King Willem II was afraid for his position after the 1848 revolutions in France and Germany and he asked the liberal Johan Thorbebecke to lead a committee to draft a new constitution. In exchange for losing power the monarch was made immune for prosecution (and still is).