Maybe this will help explain why the new webmail will not work on TB3. When TB5 came out they made major changes to TB's engine (program) that required a re-write of the webmail xpi so it would work with it. Since then Mozilla has made changes to engine each time they have release a new new of TB forcing changes to the webmail add-on just to be compatible with the new versions. Each release starting with TB5 requires a different version to be rewritten. This will most likely be the same with new TB10 that is already in the pipeline.
You can this version of the hotmail add-on that has a modified install file and see if it works for you. https://docs.google.com/open?id=0B-g4TpKUJnh7MGQ5OTE1NjgtM2UyMy00YzkxLTljNmYtYjY5NTgxZDVjMmRk
