What purpose does the task list serve? Is it for document approval? If so, you could use content approval instead, this would remove the need for a separate task list. Here is a good instruction from a blogger I follow: How to approve documents without building a workflow. You can also customise document approval to have several approvers (layers of management or approvers from different departments).
I realise I am making some assumptions as to the purpose of your task list. If your task list is to track and follow up tasks i.e. nothing to do with Document Approval, then could consider creating a custom workflow using Sharepoint Designer (it is free to download from MS). The workflow could be used to create tasks in your task list and can be configured to copy over certain fields/properties from your document library.
Updated - (Connections)
You could try this.
First - edit one of your views on the Document Library and add a web part to display your Task List on the same page. This will result in both your Document Library & Task List being displayed on the same page. Next you'll need to filter them -> imagine your Doc Library is above and Task List is below.
Next - add a Connection. Edit page -> edit a web part e.g. Document Library. Select "Connections" (see pictures). You can 'hook up' a connection between a field which is found in both the Doc Library and the Task List.
In Picture 2 you can define which fields are used - what you're doing here is configuring a filter between the Doc Library and the Task list.
In Picture 3 you'll see there is a new 'Select' field. If you test this field, you might find that you need to click select on the Document Library (to select the Project), then the Task list will filter on this.
You might find it tricky to configure the connection, it does work, I've found that it sometimes takes a while to find the correct field to create the connection.
If it does not work, try this: use the Select field AND filter on a column in your Project name as well.