Hamster chewed on my bathroom door, what can I do to fix it?

Is your issue solely an aesthetic one, ie you don't like the appearance of the little nibbler marks or is there (which I doubt) actual structural damage to the door? Chances of you needing to replace the door are very, very small.

If the indentations are of any real depth, you can get some wood putty (assuming the door is wood) from any hardware store and use a trowel-like tool to press the putty into the gnaw marks. Wait for that to dry, and if it's a different color than the door, paint over it using the same color, and that should do it. I'm sure any hardware store employee would be happy to answer any specific questions regarding wood putties and paints in the store.

Flushing the toilet might make the hamster less prone to try to escape, but that us purely circumstantial.

you could remove her teeth. just kidding!

After you repair the door, repair the cage. They are clever little devils (like gerbils) but can't pick locks, yet, I don't think.

brown marker works great