I think it makes you pay when you unequip.
This is a new pet system they recently implemented. The message that's supposed to warn you isn't right yet. (It says "Lock Seal double check tip Msg 5427" instead of a warning)
Anyhow, to use your pet to pick up stuff, just click battle.
To make it regain durability, feed it by placing the pet food in the bottom right box and clicking feed.
All of this is in the Pet Menu, there is a new button to click in the main menu thing with the item mall, bag and stuff. Alternatively, you can set a hotkey for the pet menu, the default is supposed to be N, but when I first checked it wasn't anything. If you need info, you should check the [?] button the pet menu has.
As for the nametag, there is a button "Rename" to use it in the pet menu