The push to start won't let the vehicle start if the battery voltage is below 12v I think. So, if the battery is at 11.5, all the interior stuff will work, but the starter won't engage when commanded.
You also need more info to go on. Do as suggested and get the code read with a cheap OBD2 scanner. Get a multimeter and check the battery voltage as it sits now. If it's below 12v, try charging it, or jump starting it with another vehicle.
The shaking on startup could have been a sticky lifter, or it could have been some carbon buildup that broke off and held one of the valves partially open until it was crumbled up and digested.
More homework needs to be done vs getting out the jump to conclusions mat....