design periods

Does porsche design the cars the volkswagen group makes?
Porsche is actually an engineering firm, that went into making cars as a sideline. The concept was the company should be run by engineers not by businessmen who were doing what was necessary to increase profits to please the shareholders. As far as I know they have never tried to build a large factory that would crank out thousands of cars, that would have a price low enough to compete with the really popular cars in the market place. I Think the bodies are made by Karmann coachwork and many of the other parts are bought from Volkswagen and various other suppliers. If someone has enough money they hire Porsche to design a car for them. John DeLorean was about to hire Porsche to design his car, but they refused to do the job within the period of time in which he needed the car to be designed. Many other companies have their cars designed by porsche, but this is kept secret. The engineers who worked for volkswagen may have improved on Ferdinand Porches original design, but they failed
If you go back to the time before the car company existed, there was a consulting firm owned by Porsche that did engineering projects for car companies, governments, etc. After WW2, when the Porsche car company was founded, the former consulting business more of less stopped (though it would resume in other ways years later). As Porsche became a mature company, they often maintained a much larger engineering department than would be normal for the size of their company (however, with recent cutbacks in the last 5 years or so, this is probably no longer true). The large engineering staff allowed them to pursue many production based projects, as well as race projects. To justify the large staff at non-peak times, they would do outside engineering for most all companies (Harley Davidson is probably most notable, which is why they classic Harley has such a similar looking piston/cylinder and similar engine note to the Porsche; large parts of the BMW X5… etc).
Porsche have almost certianly done work for VW, though most like more in line with the design of a door handle, and airbox, etc. Not something in the area of a whole car. They were contracted to design a sports car for Audi in the early 70s, and they did… though Audi would later decide they did not want the completed car (which Porsche then marketed as the Porsche 924).
In the early days, like many European manufacturers, Porsche outsourced their coach making to other companies like Recaro and Karmann… however, this ended when Porsche purchased Recaro’s Coachwork business (and Recaro retained only their seat making business for themselves). In fact, Porsche would at times take a similar approach to their production creating plants that exceeded their capacity needs and producting cars for some other German makes when it was not in use for Porsche vehicles (mainly happened in the 1990s when Porsche cars were not selling at expected levels). To be clear, since Porsche purchased Recaro, they have operated their own factory and built their own coachwork (and they do have a few large factories).
As for DeLorean… Porsche was indeed the first company contacted in regards to building his car. A set of specs were submitted, and Porscche responded that it could be done, but the price they quoted to do so in the planned time frame left John DeLorean with sticker shock as it was about 10 times greater than the planned budget. DeLorean then moved to Lotus, which accepted the project. Shortly after starting the project, the specs were dramatically scaled back to meet the budget and schedule. Porsche did not refuse this project, rather DeLorean refused to pay the amount required to contract Porsche to do this work.