It happens when the cold storage is warm. If the cold wallet passphrase or the private keys were not printed on paper and were accessible to a networked computer in any fashion then it would not be considered as a cold wallet to me.
I suppose that could be. It's not really a cold wallet if scripts from networked machines automatically pay "hotter" wallets when they run low.
Exactly, the keyword "automatically" demands programmatic access to the secrets, that effectively makes it just another hot wallet.