Ok, it is good. Thanks.
How can I generate a working private key manually? And how can I calculate the public key from it?
Manually as in by hand it is next to impossible. The math in ECDSA is not trivial and any error would result in a permanent loss of funds.
You can however use
http://bitaddress.org. You can (and should) save the webpage and use it offline. You can even created encrypted (password protected) paper wallets all ready to print (print two copies to put in different secure locations to avoid loss of key).