So here's a question. Say William becomes King. Kate is now the Queen? If so and William dies, does she remain the Queen as long as she doesn't remarry?
Kate will be Queen when William becomes King. If she survives William, and one of their children inherits the throne, she would then be called the Queen Mother. If for some reason they have no children, or none of those children inherit the throne, she would be a queen dowager.
Basically, she only gets the title Queen through her husband, but its hers for life unless they divorce (and even then, if they happened to divorce after he became King, she might get to keep the title, a la Princess Diana).
You have to keep in mind, also, that there are different types of queens. The wife of a king is a queen consort. The queen who holds power in her own right is a queen regnant. The widow of a king is a queen dowager. The widow of a king whose child is on the throne is a queen mother. All of those titles include the word "queen" somewhere, but only the wife of a reigning monarch or the monarch herself is ever the Queen.
And none of these queens are actually ruling queens anyway. They have certain ceremonial duties (like opening parliament, right?) but no actual governmental power?
That depends. In the UK, I don't believe any monarch has any actual authority in government, but a queen regnant has exactly the same role as a king. Queen Elizabeth II is a queen regnant -- she holds the title in her own right, rather than through her husband. Which is why Prince Phillip is, officially, a prince consort and not a king.
If the monarchy still had a governing role, then a queen regnant would absolutely have governmental power. Queen Victoria, for instance, was the reigning monarch, and did have an active role in government.
kareng57, Wikipedia agrees with you about why we call Elizabeth II's mother "The Queen Mother." Other dowager queens would have just gone by "Queen Mary" (or whatever), but their official titles at that point would have had the queen mother bit tacked on. What I found interesting is that, for about a year, Great Britain had three
queens: Queen Mary (wife of George V), the Queen Mother (wife of George VI), and Queen Elizabeth II.