If it is about "apple corps" business, then you face potential TM issues. If it is about apple pies, then you are fine.
If the name has been copyrighted in the past before you get yours online, technically yes they have the right to take you to court. However, if your content is nothing to do with that site, then that's okay. They don't own the internet. ~ Slutz ~
This has nothing to do with a "copyright" - names aren't copyrighted. While usage plays a key role in trademark infringement - famous marks, especially coined terms "google" "xerox" have broad protection where any use of the mark is going to be infringement. It has nothing to do with owning the internet, it has to do with owning the rights to a mark - and those rights extend past the internet.