I hope you replaced the crankshaft and camshaft sensors with Bosch parts as other brands have been found to be lacking. I am not saying that is your problem though, but the plugs do not fit the sensors properly on some makes.
The remote entry feature has nothing to do with starting the car. You mention the SKREEM but did not say definitely that it did not start only three times before you reset it by disconnecting the battery, but I assume that is not the case.
Have you done anything to the car prior to the problem lately?