Hi A.,
There are a lot of good answers here already. Basically, your daughter needs a reliable source of calcium, B vitamins, protein, some carbohydrates, and healthy fats for good growth and development (among a host of other nutrients and minerals). Everything she needs can be supplied by a healthy diet, with or without dairy, and with or without animal products. It is a LOT more difficult for a toddler to gets the fat and correct protein complement she needs through a purely vegetarian diet, however.
While green leafy vegetables are a great source of calcium, an easier source (in terms of volume that needs to be eaten to ingest the necessary amount) is dairy products, which can include cheeses, yogurt, kefir, etc. Calcium can also be derived from eating small-boned fish (bones and all), such as sardines, but I'd be a little concerned about these presenting a choking hazard in a 1-year old. But, the greens are still good if that's all she'll eat! What I DON'T think is a good idea is adding any kind of sweetener to her milk (like instant breakfast, strawberry or chocolate syrup, etc.) to get her to drink it. Childhood obesity is a real problem in this country and a significant health risk. Why potentially start your child down that road by getting her used to things having to be sweet in order to be palatable?
As far as protein is concerned, dairy is also an excellent source, as are beans, lentils, tofu, and most whole grains (and fish if she'll eat it). It's important to eat a wide variety of these to get the full amino-acid complement the body needs to build it's own proteins, as some are lacking in beans while others are lacking in grains. An excellent grain with a high protein content is quinoa, which can be made as easily as rice and seasoned just about any way you want to for lots of variation.
You also raised a question about raw vs. pasteurized milk. While there's a certain faction in this country that seem to think pasteurized milk products destroy important nutrients, this has never been scientifically proven (in fact, the opposite is true). On the other hand, eating raw dairy products puts your daughter at risk of contracting listeriosis, which is a serious and sometimes fatal bacterial infection. Yogurts and kefirs offer good compromises in this department, as the process of making them and the types of bacterial cultures used are actually a form of "controlled spoilage", and generally safe. I'd still not risk using unpasteurized forms for a toddler, however, and pregnant women should NEVER eat raw dairy foods as these can cause fetal death and neonatal sepsis.