Can an Earl marry a Princess who is to become a Queen who will rule a kingdom?

Yes. The princess still is a princess but also a countess. There is a real world example, the current UK's Queen Elizabeth and her sister. When then Princess Elizabeth married Philip, Duke of Edinburgh, she was titled, Princess Elizabeth,the Duchess of Edinburgh. When her sister Margaret married Antony Armstrong-Jones the Earl of Snowdon, her titles were the Princess Margaret,Countess of Snowdon. Both women retained their royal titles and in Philip's case, he was given a UK royal title to replace his foreign titles. The Earl assists his wife, just like Prince Philip, Duke of Edinburgh does, but since this is in Medieval times, there might be more of a military role for the husband. The Earl may become a Consort. Perhaps he will be made Prince Consort. If really is up to you if your character wants to make her husband King Consort. An historical queen who did make her husband such was Mary I of England.

Why yes of course, dear girl. As a matter of fact, it would be a very splendid thing. Why not move the plot of your novel to the 20th century? Perhaps the 7th Earl of Lucan and the wonderfully attractive Princess Elizabeth might tie the knot, so to speak?

First off: the idea of a king giving up his throne to his son, but getting the throne back when the son dies, is utterly foreign to any medieval monarchy. European monarchy just doesn't work like that, and never has. Of course you can invent a fantasy world in which monarchy does work that way - but in that case you can make up your own rules about everything else too, and it's pointless to ask anyone on this board about it. But if we're talking realistic medieval royalty here, then the princess is not going to be allowed to decide who she wants to marry: that decision will be taken by the present king. It will be a crucially important political decision, and love won't come into it. Basically, the options are: 1. to marry her to a foreign prince or king, and thus create an alliance with that other kingdom, or; 2. If that doesn't look like a good idea - e.g. there are two possible marriage alliances and choosing one will probably set off a war with the other - a husband could be chosen for her from the nobles of her own kingdom. That carries its own set of problems: it's demeaning for her to marry one of her inferiors, and the jealousy of all the nobles who don't get chosen could start a rebellion or civil war. But it could happen. In the Middle Ages a queen's husband was automatically king. This was partly because a monarch was required to be a leader in battle, a job a woman couldn't do; but also because a wife was legally (and in religious theory too) subject to the authority of her husband, and had to obey him. A Queen could only be the boss if she had no husband at all.

The Earl can marry the princes. She would become a countess, sharing his title as a courtesy. He would get nothing from her. The King may grant him a higher title upon the marriage - probably Duke. When the King dies, the princess becomes Queen. Nothing happens with the Earl - he remains whatever he was. As happened with Elizabeth and Phillip, at some point, she may well grant him a title, or make him a prince. But, he would only be a consort - she would reign as Queen Regent.

Why would the old King's son become King? surely he would only become King once his father was killed? The Princess would only become Queen if there were no other male relatives, cousins or whatever to pass the crown down to. It would also be unlikely that she would be able to marry who she wants, there would likely be some sort of political reason for her marriage. The Earl wouldn't become King, he would be a Duke once they were married and she was Queen. She might be a Countess if they married before she became Queen, but once she was Queen then that would be her title.