Seat are part of VW group, as are Audi, Skoda, Lamborghini, Bentley and Bugatti (they still own the Auto Union and NSU names but don't use them). Porsche, although the smaller company, owns more than 50% of VW shares which should eventually lead to a merger between the two companies. One potential fly in the ointment is Porsche have borrowed money from the Middle East to fund their share purchase in VW and I think we all know of the problems there at the moment.
In terms of motor sport, VW go rallying with Skoda, run Seat cars in the WTCC and Audi in the DTM, supply engines in a couple of F3 series and, of course, run diesel cars in the Le Man series. I'm not sure whether any of this qualifies them to make F1 engines, perhaps they will use the Lambo or Bugatti names but if they are after publicity I would presume they would use the VW brand. One problem with all of this of course is would Ferrari, Mercedes or Renault want to run in F1 with an engine not made by them, some how I think not.