Just click the pet so it is highlighted, click the other tab and click an empty stall. The pet will automatically be moved into that stall in the new tab.

Ooh ok Lolz thanks

If you hover your mouse over the picture it tells you what it is. But just in case you're not using a computer:

Map - exploration item (can be found/bought as you explore the regions)
Gold - merchant item (can be bought from merchants in the town)
Girl (It's actually Faiona's head I think, one of the admins) - forum game item (can only be won from an official forum game)
Sun - holiday event item (can be found during festival events, mostly boss item drops/festival shop items)
Diamond - avatar shop item (can be bought using diamonds in the avatar shop)
Calendar - monthly avatar item (can be bought using diamonds in the avatar shop under monthly items. The items are changed monthly, so you can only buy what is being sold that month. Not sure if they bring back past items)

Oh ok. Thanks

Also why is there a notification that says "if you are alive push the diamond"?? Why is that there when I nurture eggs?

It's for the system to check if you're a human. Don't really know much about it, but it prevents people from using bots/software/whatever to help them "play" the game without actually being online. Something like that. =/

It's like how when you sign up for a lot of things online they ask you to verify or type in a string of words/numbers.

Huh. That's strange, using robots to "play" but not the person. XD lolz how is that fun X3