You're welcome.
-When and how is the old iPhone deactivated? Does it automatically happen when the new phone is connected to my MacBook and restored from backup?
Your existing iPhone will be deactivated the moment your new iPhone is activated.
-After it's deactivated, if I erase all contents and settings from the old iPhone, will it go back to the original software version?
The erase all content and settings options does not touch the firmware. It is a secure erase for all content and settings only. The existing firmware installed is not touched or changed.
-After erasing all contents and settings from the old iPhone, when my daughter connects it to iTunes in her Windows machine (which won't recognize that phone), is it not going to ask her to activate the phone or for a phone number?
iTunes will recognize the iPhone - just as when an iPod is connected to iTunes. Unless you insert a new non-activated SIM card in the iPhone, she will not be prompted to re-activate the iPhone with iTunes activation process.
-Do I always have to keep Airplane mode on to be able to use it as an iPod, even that it will be already deactivated?
I do with my first generation iPhone when used as an iPod Touch. This turns off all iPhone radios - cell/cellular data, bluetooth, and wi-fi. You can follow this by turning on wi-fi access. With the other radios off, this will conserve battery consumption, and the other radios - cell/cellular data and bluetooth won't be used anyway with a non-activated iPhone. If firmware update 3.0 is installed on this iPhone, she may want to turn Airplane mode off in order to turn bluetooth access on if she wants to pair with another iPhone for playing a game, etc. If not, I would leave Airplane mode enabled with wi-fi access turned on for internet access via wi-fi just as with an iPod Touch.