Changing the name (or location) of the TextEdit app, or any of your other Apple apps as well as some third-party apps, will cause it to stop being updated and can cause apps to stop functioning altogether. Apple didn't make it difficult to rename such things just to mess with you! There are reasons.
As for the Trash, the only way you could rename that is by hacking a system somewhere, which is never a good idea. You could very easily break something in the process.
Just leave the names alone. There are much better and safer methods of self-expression.