ER Modeling; multiplicity basics – one to many or many to many

Why must the employees working on a project strictly belong to the same division that owns the project? That sounds more like a business rule that you can leave up to your application to enforce rather than something that requires DRI.

But let's say you need to enforce this using DRI. You said you have a many-to-many table associating employees with projects. Assuming you are using SQL Server, you could expand this table as follows:

CREATE TABLE dbo.Employee_Project (
     ProjectID            INT
   , ProjectDivisionID    INT

   , EmployeeID           INT
   , EmployeeDivisionID   INT

   , Hours                INT

   , CONSTRAINT FK_Project FOREIGN KEY (ProjectID, ProjectDivisionID) 
        REFRENCES dbo.Project(ProjectID, DivisionID)

   , CONSTRAINT FK_Employee FOREIGN KEY (EmployeeID, EmployeeDivisionID) 
        REFERENCES dbo.Employee(EmployeeID, DivisionID)

   , CONSTRAINT CK_EmployeeProjectSameDivision CHECK (ProjectDivisionID = EmployeeDivisionID)
);

This is what's going on here:

• We have two composite foreign keys here, one for Employee and one for Project, so we can have the DivisionID for both entities in the same table and guaranteed by DRI to be correct.
• These composite foreign keys require a UNIQUE index on the referenced tables. So in addition to your primary keys on EmployeeID and ProjectID in those tables, you'll need unique keys on (EmployeeID, DivisionID) and (ProjectID, DivisionID).
• The CHECK constraint guarantees that an employee can only be assigned to a project owned by the same division.

If you want to see another example of this design pattern, here's the answer I gave to a design problem that had similar requirements.

One problem with your proposed design is that instead of simple JOINs you have to run dynamic SQL every time you want to see the actual value of any entities. Furthermore, you will have complicated procedures (or triggers) for inserting data into ProjectContents.

Stick with the ProjectEntity1, ProjectEntity2, ... solution if you can have more that one EntityX coupled to a certain Project. If this is not the case (that is, a Project can have maximum one of each Entities), you can create a table like this:

ProjectID | Entity1ID | ... | EntityNID

In either case, your JOINs will be relatively numerous (not that 10 JOINs in a query are a lot) but otherwise simple. And, lastly, occasional maintenance (like adding new entities) will also be simple.