That is an awesome idea. There is an addon called Thunderbrowse that allows you to open websites in Thunderbird. If you open it to the Slim website http://www.toodledo.com/slim , it is basically like having the FF sidebar. Ultimately, some way of syncing Toodledo tasks with the Lightning addon is my dream.
If you have trouble logging in to Toodledo with Thunderbrowse, you will need this addon also: https://addons.mozilla.org/en-US/thunderbird/addon/5207
It controls/allows cookie usage. Some people don't seem to need to use it, but I do. I guess it depends on your version of TB or something, but I'm not sure.