Managed to get the following working, in the following ways:

• Sunbird: exported old data to an ICS file on Windows, then uninstalled Sunbird on Ubuntu, deleted all profiles, reinstalled, then ran it and imported the data. Importing took a while, but it seems to work now.
• Firefox extensions: Added Ubuntu extensions (most of what I need is there).
• Firefox Java: A hard one. This required me to download a snapshot from here (jre-6u12-ea-bin-b03-linux-amd64-22_dec_2008.bin), then unravel it into /opt/, and then point Firefox at its javaws to handle JNLP files.
• Got WinSCP installed under Wine (it's still the best tool for that job).
• Began rebuilding my bookmarks and bookmarklets (they needed prining anyway).