How do you remove silly putty?
You could try freezing the area with ice and then chipping away the Silly Putty. However, it's my experience that once it's set in clothes and carpet, it's pretty much permanent.
No, it does not harden! It is always playable, even if you forget it on your desktop. The only way to harden silly putty would be to expose it to extremely cold temperatures. For example, Silly Putty hardens when put into liquid Nitrogen. It will harden the putty, but it will regain its texture as soon as it warms up.