Well if the check engine light is on it means there is a fault in the system other than bad fuel and battery. I hope you have another set of wheels to carry you around as you shouldn't be driving this car until you've got it figured out! Thats why I was asking about the OBDII meter since if you read the code it'll identify the source of the problem. But I understand not everyone would've an OBDII meter lying around their house!
Just sit tight and wait for the mechanic or if the situation allows take it to the dealer's shop so they can hook it up to the system and find out the fault. Even if the mechanic comes to your house and he can't hook up the computer it won't help you and you'll still have to go the dealer's shop. If you have the warranty you have nothing to worry about. It should be taken care of at the dealership.
If you hear the Kherr kherr sound, it means the battery is fine and the starter is trying to do the job but the engine won't turn over. So it can be anything from a loose hose, a vaccum leak or clogged fuel pump and also a broken timing belt! (Yep i've seen a broken timing belt in a Toyota with less then 80 miles on it). You just can't tell by looking at the engine you need a code print out or enough knowledge to check the systems one by one. If you pull out a spark plug wire and hold it close to a good ground while starting - if it gives a good spark than rest assured your battery, ignition system are fine. Its either a vaccum leak or clogged fuel system. I'm saying clogged system again and again since you mentioned the car has been stationary for some time and with winter there just might be some moisture built up in the system DUE to a leak! You are lucky its parked in your home and you're not stranded 300 KMs from your home on the side of the road in the middle of the night!
While waiting - do not blame corolla, Toyota, your dealer or why you made the decision to buy the car. It can happen anytime, to anyone and in any car. Its just bad luck it happened to you at this particular time.