If you copy and redistribute the code as-is, on a website called, let's say opensourcesoftware.com, nobody can stop you.

If you copy and redistribute the code on a website called thisistheofficalsitefornotepadplusplus.com, because the ownership of the name Notepad++ is separate from the code.

If you copy and change the code before redistributing it, they can say you're not allowed to use the name Notepad++ and must change that part of the code. You asked a very pertinent question with "Does the GPL say anything at all about trademarks?" It does not, because it has nothing to do with copyright. It does say if you modify the source code, you must make available the your modifications under the same license.