This raises a couple of our favourite tips too!
1) Disney is very good about providing ice water for free at its food venues - just ask!
2) If you have access to a freezer where you are staying, try bringing the "bag" type water bottles (we have Vapur). Freeze them overnight (but don't overfill first) and then use them to keep your lunch cold during the morning. We usually go in March so they are mostly frozen still by noon, but hang them outside your pack and the water will thaw from the afternoon sun Used with a cold hard bottle for morning, and tip one above for refills, we have ready cold water throughout the day.
And yes, we bring full but compact lunches to the park every day and have never had a problem bringing them in. Each person in our party has a MEC sling pack that easily holds lunch, pancho's, sweaters, etc and slips around the front for easy boarding of the rides.