I would consider a referral to a dietician or nutritionist - most physicians know very little about nutrition, actually. The good ones admit it, but most have never even had a class in it. For example, the skim milk argument has recently come under question - our obsession with low-fat foods has resulted in us not being satisfied by fats that take a long time to digest. So we're hungry again too soon and we go for the next snack/meal.
Don't avoid fats - she needs the healthy fats, which you are already giving her in other dairy (cheese, yogurt) but try adding nuts, avocado, and olive oil. For example, make your own chicken nuggets or fingers by cutting up chicken breasts and dipping in egg and then wheat germ (or a wheat germ/flax meal mix), then quick-frying in olive oil. I just crisp up the outsides and then put them on a rack on a baking pan and finish them in the oven. Sweet potato or even white potato oven "fries" (or mix/match the potatoes) drizzled with olive oil and maybe a favorite herb (we like dill) make a nice accompaniment and satisfy the "fries" craving. You can make her carrots and broccoli the same way.
For dessert/snack, try putting plain cocoa powder right in some vanilla or plain Greek yogurt. It's a creamy alternative to ice cream, and the cocoa only has about 5 calories in it. If she wants soda for a treat, you can use plain seltzer or add in a squeeze of lime/lemon.
I make a smoothie every morning with my nutrition shake - I add in frozen fruit (cheaper than fresh and perfect when you're grinding something up in the blender) and sometimes spinach or kale (just grab some from the bag of frozen stuff in the freezer unless you have it fresh on hand). It's a great way to use up something that's a little soft but not spoiled - the soft strawberry, soft peppers, squishy grapes, etc. Put in some almonds for the healthy fats. You can add the plain cocoa there too.
You can create your own pizza with store-bought dough, and get her involved in making/helping. If you have a farmer's market near you, take her to help choose some new things. There are great alternatives and ways to "hide" things - eggplant slices baked/grilled and then rolled up like lasagna rolls, zucchini grated into spaghetti sauce, spaghetti squash instead of pasta, and so on.
What concerns me is that her tummy hurts after she eats - sounds to me like she is taking in too much at one sitting, and doesn't really feel satisfied. That's where those healthy fats come in. Some peanut butter in celery sticks is a good snack too and great for lunch boxes. Be sure she's getting protein at breakfast - eggs are an excellent source, far better than cereal and milk even if it's skim.
As for your remark to her, I think you have to be honest about it - you said it without thinking, it was a throw-back to your life as a heavy kid. It's a good example of parents not being perfect, and of repeating what was done to them. It's also a way to open up an ongoing discussion (over the next 10 years) about the body images we are given by others, the media, the fashion industry (which has notoriously low sizes, by the way), TV shows, and so on. Look at pictures of Marilyn Monroe, when a size 12 was normal! What has happened to us?
Let her know that your body changed a lot and hers will too. Keep up the activity and try to do some things together.
It sounds like your daughter has two other issues that need attention - a very late bedtime after a late dinner, and a very overweight father. I don't know how much you can do about those things. If he can get walking with her, and if they both get away from the TV and take hikes or throw a ball around or plant a garden, it will be a great experience. Not all exercise needs to be in structured classes.