Hi
We have travelled with our children a fair amount - (the oldest is now nearly 2). I have always taken his sippy cup with water in it plus a bottle of water and small carton of juice and had no problems. I know what you mean about grapes. I have taken them in a multiple small boxes (portion size) to stop them getting squashed and eaten all at once!!! I have had to dispose of any uneaten grapes immediately on landing due to the restrictions of bringing fresh food into the particular countries we have been to.
Our 2 year old prefers walking so has walked through the scanners between myself and husband, but I have seen others being carried so whatever works and see what happens.
Enjoy your trip.