Question: I would find it more interesting/organized if there was a theme to follow for the featured images each week. For example, this week post only pictures of cars, flowers, landscape or whatever and change the theme every week.

Answer: 
Yes! This is a good idea. It would add some interest and focus to the pictures of the week contest.
The contest idea, this is similar to many photo sites I am familiar with in the past. 

Personally I don't typically have time to go out and shoot the theme every week, so I would likely participate less if ever.

It does have some merit to expand your portfolio or challenge one, but it's just not my favorite way to shoot or think about shooting.