I did this many times, the chain is not downloaded fully and after a while, error occurs always on start.

Can you PM tonych the error, etc... so that he can help?

Is it possible to send everything in the wallet, that is not up to date (maybe state of two month ago), to another new created wallet, even without displaying the correct current amount? On the faulty state that still works, it is ~5GB, but the address holds ~6GB to the time of now. I'd like to transfer the full amount ~6GB byte and 12GB blckbytes to new wallet.

I believe that you need a fully in sync client to send Byteballs or Blackbytes

Is lightwallet better? Does the lightwallet hold blackbytes?

I have not used the lightwallet but I believe it can handle blackbytes, unfortunately your issue is getting your coins out of the current wallet. Again try PM'ing Tony about this.