Hi Applesnail welcome to the forum.
Axolotls do indeed eat snails that fit into their mouth. I have an axolotl who loves snails and actively "hunts them".
There are mixed feelings about apple snails and axolotls living together.
Apple snails (as you probably know)have a hard door on their foot and can close up tightly, should an axolotl have its face, foot, or tail near the snail when it closes it could very easily be caught in the snail closing up.
There is a "disaster story" about that here:
http://www.caudata.org/cc/articles/Mixing_disasters.shtml
Scroll down to the axolotl and apple snail mixing story .
It is true apple snails will eat meat that includes the dead and the dieing and possibly the slow.
There is a great site about apple snails here:
http://www.applesnail.net/
Check out the FAQ
I have apple snails but I do not keep them in the axolotl tank. I can not recommend putting snails in your axolotl tank as any number of things could happen...or nothing at all. There is risk with putting any 2 species together I will always vote in favor of the axolotls safety.