Too much milk will cause iron deficiency and anemia.
My daughters are allergic to cows milk protein. It is a different allergy than lactose intolerance. Google the specifics, because the dietary problems/needs are different. They drink goat milk instead. I think Mullenburg is the normal goat milk brand in most grocery stores. We buy the whole milk (purple carton), not skim.
Lactose intolerance is typically a life long allergy (your body does not make and lacks the enzyme to breakdown lactose). A cows milk protein allergy is common among infants and young children and usually is outgrown by age 3-5.
Yes, cows milk allergies can cause either constipation or loose bowels, congestion, eczema, poor weight gain/growth, etc.
Your pediatrician will tell you to get off bottles sooner than cutting the milk intake. Start introducing sippy cups. We had success with the Nuk sippy cups (more than the other brands).
You should target about 28-32 oz (of milk) initially after the transition (age 12 months) and then over the next 4-6 months, decrease it to 24-28 oz. By the time she is 18-24 months, it should be 16-18 oz. After age 2, target 16 oz of milk per day.
Easiest way is to introduce a sippy cup of water during snacks and meals, and use smaller sippy cups (5 oz) instead of relying on 8 oz bottles. That way, when its time to decrease milk, you can easily omit a little cup here and there without a huge upset in the routine.
My daughter is currently 14 months. I prepare six 5 oz sippy cups for her and often she doesn't drink the 6th one. If not, I save that for the next day. She tends to drink 6oz as soon as she wakes up (while I am changing her), 5oz at 9:30, water at lunch(11:30-12pm), 5oz after nap, water at afternoon snack, 5 oz an hour after snack, sometimes 5oz around 5:00-6:00, water at dinner, 6 oz at bedtime. The 5:00 one will be the first one to go... Then, the one first thing in the morning will be cut; she'll fill up her tummy on breakfast instead. Then she'll lose the 5 oz an hour after snack. Again, you'll end up replacing milk with actual food as the child gets closer to age 2. When our first daughter was age 2, she drank 8oz of yogurt smoothie in the morning and another 8oz when she got home from school; water with all meals and I offered apples as a nighttime snack.
Good luck!