Mozilla Lightning: Linking emails with tasks

Use Google Tasks Sync for Mozilla Thunderbird. It requires Lightning for calendar support.

There is no integrated search as far as I know. However, there is a rather good search function in Lightning, you can access it under the main menu > Events and Tasks > Find events.

A search bar along with a list of events will appear on top of your main calendar view. You can filter the search criteria on a time basis and expand the events information (and sort the events accordingly) by adding columns to the list as you would in the message list pane.