There are a few separate issues going on here.
1. You have not yet enabled chatter on the task object. Go to setup - Feed Tracking and add the Task object.
2. You are building a home page component, so naturally you don’t want the standard salesforce header. But Chatter depends on a javascript file that is loaded with the header. This means the page include mechanism described above will not work. So in this case you have to go the “template with an iframe route”. In addition, we had to create a different “chatter VF page” in your org that removed the header but still called the right stylesheet files. This has been done.
3. I created an account button on the tasks for one of your tabs. Check it out. I think you will be able to propogate the concepts to other areas.
