How seriously do the admins take this rule?
Or what is defined as owning? Is it enough to have it on my hard drive?

Forget uploading the scene releases bcoz it is done by the tracker BOT. As far as remaining content is concerned no one cares i think as i have seen the same content on many trackers and it is uploaded by the staff. Anyways its better to read the rules and FAQ of the tracker first and if you have any doubts go to the IRC channel and ask the staff.

I wouldn't worry too much about it. Some trackers reward you for user releases but it doesn't matter where you get the content from. Just be careful because some trackers do have 'exclusive' releases that they don't want ending up on other trackers.

And what about legal rights of the file? What is meant with that?

If the material is copyrighted then you are not the legal right holder. It would be technically illegal to upload such files. But there are always ways to protect yourself to some degree.

This is just a way for tracker owners to "protect" themselves against legal problems. Just forget about it because it's not helping anyone.