I always wondered if Gertrude wasn’t protecting Hamlet by marrying Claudius. Lots of less-powerful would be heirs were bumped off by more powerful rivals (and, indeed, in the end, Claudius decides to do just that to Hamlet). Perhaps she was coerced into the marriage to protect Hamlet from Claudius. Or maybe just in over her head. I always felt rather sorry for her, myself.
Except marry him and beg his mercy for the life of her son. It was the good move on her part.
Stranger
There are many female characters in Shakespeare’s plays who are stronger than Gertrude, and influence events around them in spite of lower status. And the strongest person politically in Shakespeare’s England, from his birth to when Hamlet was first published, was a woman. So, in another story, the widow of a deceased king could have taken other action, including going into exile to rally forces around her son. But it is pretty likely that Claudius sweet-talked Gertrude into supporting him as king in the election – which might have been simply by marrying him.
From studying it in school, finding out his age was one of those tricky but surprising things. He’s actually thirty. It’s mentioned in a really obscure way (some random thing happened thirty years ago, which happened to be the date of his birth–think it was the graveyard scene).