how can I adhere it to the foam without distorting it?
You can use adhesive, Velcro, and glue. Just make sure the support backplane is flat enough.
It says the back is also a mirror film, does that mean that this is a one-way mirror?
This is your standard reflective mirror. It is not a see-through one side, reflect the other "security glass" -style mirror.
I was interested in using this mirror for a child's vanity. Would this type of mirror work for this project?
Yes, it's just as clear as a regular mirror would be.
True on the sharp corners, but with the corner clips you can purchase to attach it with, it shouldn't be a problem. You can also sand it down.
It should. It's not ultra shiny, but it won't shatter and it reflects.
Is the reflection as clear as a typical household mirror would be?
When brand new yes, it is surprisingly clear. But I imagine with scratches that a glass mirror would not get, it would become less clear.
It is pretty clear. You can tell it isn't glass, but it isn't "fun house". I got it for my infant son to use on his play mat, but I could easily see buying one to stick up in like a locker room situation to do my hair or make up. I hope this helps answer your question.