You are right, there was a problem with the cookie domain settings. Although I modified it as it is suggested on the JFusion board (with a "." (dot) in the front of the domain name), it did not work. I entered the values like this: .mysite.com/forum and .mysite.com/joomla. That is not good, the correct setting is: .mysite.com. With this setting it works like a charm.
So go ahead, drop your Fireboard and convert to phpBB3. It is not a one-click job, but doable.
phpBB3 works with jooomla 1.5.8, and using JFusion 1.0.8 it is no problem that the passwords cannot be overtaken during the forum conversion.