There are differences in function, not just in purpose.
Kid's Corner is more convenient to access. You enter Kid's Corner by swiping left from the lock screen. To switch back, don't swipe left at the lock screen. Apps Corner is accessed by pressing the Launch button in Settings > Apps Corner. You return to the regular profile by holding ...
Kids corner is a special area where you can have your kids access only the shops they you want them to have. It is always accessible. The rest of the phone is behind a pin.
App corner is a special area where you can have only a certain set of apps visible. You activate this mode manually. It can only be deactivated by a specific key press. It is meant to ...
No, calling (ie. the phone app) can't be added to Kids Corner in Windows Phone 8.
When setting up Kids Corner, you can add Games, Music, Videos and App you've download from the Windows Phone Store but not many of the built-in apps as the FAQ on Kids Corner mentions...
Because they can be used to make changes to your Start screen or phone
settings, Hubs ...
By design the kids corner have no PIN.
Your PIN should only be asked for when you try to log in on the non kids corner side.
The problem however is that if you have your phone connected to a domain that domain might have a group policy forcing your phone into always asking for the PIN. This will also make the phone ask for PIN inside the kids corner.
I have same problem. And I'm afraid answer on your question is no.
From MS site:
If your phone is set to lock itself automatically after a period of
inactivity, you might sometimes need to enter your lock screen
password before your child can open Kid's Corner.