As long as I see here, a pet can have a name and a title. I know how to name and also how to rename a pet if I wish so. But how can I put something as a title? If I click there, in the fields is standing "-none-" and it can´t be changed. Does that mean, that can not be done willingly? Has a pet to earn a title or something like that?

You're suppose to click the pad & pencil (same as for when you're renaming your groups) and select one -- which you'll need to make first by clicking Manage My Titles, same place as Manage My Group. (The button is Add New Title, and you can only have three, unless you buy more, which is located in the Premium Store for 15 gold -- Not sure if you can be rewarded them, and I don't remember seeing any in the Item Trades.)

Oh, thank you so much, I didn´t see that - that is very helpful. As I couldn´t find anything to manage it, i had the idea of that being a reward from an exhibiton or whatever or something like that.

Aww, I was too late to answer this. I have a guide up with pictures on how to do this because someone asked previously. I will just silently slip it in here if anyone else comes along and would like to see. Here it is.