Yes, it is free to download and free to use.
Right now during beta testing, GNU/Linux and Windows. If everything goes as planed, mobile devices will be supported as well.
Yes, this is no problem at all. On the contrary this increases the numbers of friends you can peer with, especially if at least on of the computers is directly connected to the Internet.
Copy your key ~/.whisper/keystore to the new computer you installed the Whisper software on. Login as usual. You do not need to copy any database files or so, it is handled automatic by the Whisper network. The keystore file and your password is everything needed to Whisper, since all your data is stored all over the Whisper network by you and your friends.
You don’t. Without a central server or organization, there is no authority available to restore your account. Make sure you backup your key and write down your password in a safe place, preferable before you lose it.
No, not at the moment. Today, Whisper is a prototype, and those kind of features are quite low on my list of things to-do. First the usability of Whisper stand-alone is prioritized. But feel free to add things to the WishList.
You may have re-registered the same e-mail twice. Since all your friends already knows you, this newly registered account with an already used mail address, will be treated as a identity theft, and all communications will be silently discarded. Copy the file .whisper/keystore from the computer you originally registered your Whisper account on first.
You simply close the program. Since there is no sever you are logged in to, no need to explicit logout is needed, nor possible. A program close will exit the program gracefully, closing all local files etc.
Green friends are friends you are currently peering with, i.e. have a direct connection to. This is not to be confused with friends online. Being online or not, is a much more complicated topic, since we then first must determine online with whom or what. It is perfectly possible communicating in real time with friends that are not green as well.
Whisper being a store-and-forward network, the concept of being online is not well defined.