Is it appropriate to email a professor with a simpler solution than the one she provided?

Email the professor, explain that you came up with an alternate proof, and ask if it is correct.

If you approach it this way, it should be perfectly acceptable to anyone. You get to find out if your more-clear approach is correct, and assuming it is, the professor maybe uses it in the future. If it's not correct, you get to learn where you went wrong. A good exercise for all involved.

It’s okay to ask if your proof is right and, if it is, ask if there is an advantage in presenting an apparently more complicated version of the proof.

Keep in mind that, when teaching, the technique used in solving a problem can be more important than the solution itself. It might be that this technique comes again later in the course, so having seen it once will help the second time around.

Yes, it’s fine. Professors are not snowflakes. You can communicate with them just as you would communicate with any other person. And presumably this professor is interested in the topic she is teaching and would be happy to learn about the simpler proof you found.