About the milk, not a problem. As with the usual drinks; milk, tea (hot & cold), juices, water, etc. are free and available during meals in the dining room or anytime in the buffet area.
As for Cozumel, you won't encounter any problems there. We've spent probably 2 months there over the last 10 years, with all the cruises. We even stayed there twice for a week each time.
But that's what I hate about the press - they tend to blow everything out of proportion and make people afraid of everything. The violence is in some of the border towns and a few places on the western coast. In Cancun, Cozumel, and many other tourist resorts, there are no problems. It would be tantamount to people in Europe seeing there were crimes in LA and deciding not to come to the US at all.
As for whether your daughter can stay onboard while you get off the ship, I'll tell you their "official" policy on the matter. You can leave her in Camp Carnival while you go on a Carnival shore excursion. This way should something happen, they know how to get in touch with the tour operator. If you're not on a Carnival shore excursion, they say they have no way of getting in touch with you.
Now, with that said, once you get onboard, I would go talk to them and let them know you can provide all the information they need; where you'll be going, the telephone number of the place, your cell phone number, etc. They may (or may not) be able to accommodate you.