With my newly built Cartographers Guild(I'm so happy ^^) I noticed that I keep finding the same pets, just in different positions. I know how the changing of positions work, I'm just wondering, do the pets ever change? I keep finding two gryphons, two Pumas, and one cockatrice, but it's always those pets...?

I believe you have to capture one of them to clear a place for a new beast to appear and take over its den.

They will leave on their own eventually as well. After they drink and hunt they will return home and vanish.

And different pets will appear?

I haven't done any capturing in quite some time, but as far as I know lairs can spawn any pet. Also, I think lairs up high (cave, ruins) and the island can spawn only flying pets.

I'm not quite sure if they change if you don't catch them(maybe they don't and that's why you keep seeing that particular set up?), but when I was capturing everything I saw it definietly was random beast in place of captured one. So maybe just catch everything and you surely will get some desired beasts eventually.

Lastly, remember that  there are only dragons, rocs, gryphions, pumas, unis and cocatrice in "catch wild pet". Wild symurg and throdama can be aquired at the portal. Wild lykos were aquired during winter event last year (some players still have unhatched eggs ;).

I haven't caught pets in a bit, but they do seem to "rotate' every cycle or so. I tended to place catchers ahead of the curve in areas where I think the pet may land. 

I can dig up some old guides if you would like. It's been ages since I have caught anything, my stable can't handle new pets :(

The FAQ on cacthing wild pets helped me a lot.  I recommend starting there, but then feel free to PM me if you have any more questions!