Can I drink/eat cold foods like ice or ice cream while having tonsillitis?

Absolutely! Eat any food that is comfortable. Usually cold and smooth foods like you describe go down more easily, and may even provide some soothing effect with the cold. I always tell kids with sore throats to eat what they want to and not to worry about other things for a few days. If that means an ice cream or milkshake diet for a couple of days, so be it.