What key opens a song but not a door?

What fire can cool tempers instead of heating them?

What map changes every day even when it stays on the same wall?

What paper can cut through silence without making a sound?

What tooth bites paper but never chews?

I help mistakes disappear, but I make crumbs when I work. What am I?

What hat helps a group think, even when nobody wears it?

What mountain can fit in your pocket but is too sharp to climb?

What pencil never writes, no matter how sharp it is?