In short, yes and yes. Here is an example: Cheryl's horse stumbled, throwing her to the ground. And it definitely sounds more natural with "to". However, note that stumble means: "to fall or almost fall while you are walking or running". In your sentences it was clear that stumble means "to fall", but if you wrote that:
- I stumbled.
- I tripped.
Then, the first sentence can mean that you fell as well as that you almost fell. As a result, these two sentences not necessarily have the same meaning. Though, in your sentences, you made it obvious that the result was a fall, so the words could be used interchangeably.
Student or Learner