I bought my daughter a 07 Mini Cooper S with 53,000 miles. I checked it out before buying and it made the 300 mile trip home fine. My daughter ran the car hot (to point of it shutting off) . I trailered the car home to keep from running it any more. I replace the thermostat and housing. The car is no longer running hot but...It's now very hard to start when cold and it idles rough. Took it to A/Z to run the codes and I got these; P0300, P0302, P0303, P0300, P0302, P0303, P0304. I have not detected any coolant in the oil or oil in the coolant. I plan on checking the spark plugs later today. I'm very concerned that I have a blown head gasket or worse yet a cracked head.
I need some direction.
Thanks

PS- 1st post