starter problems

Yes, it is standard to note defects with a vehicle when it comes into the shop. In a lot of cases, this leaves customers feeling like they are being scammed because they come in for one thing but something totally different is recommended. I would not say that the time to warm the engine is definitive, since the engine will be cold after hours of being driven. Especially if the vehicle was not driven for more than a few minutes. The biggest problem you face is without someone there saying they broke the car, it can be dismissed as coincidence very easily, and it could very well be. Don't drive the vehicle. The coolant is mixing into the oil. Head gasket failure on a 15 year old engine... likely.