Dear J.,
Yes and yes. Yes, there's a good possibility she could have a cow's milk allergy, and yes, I recommend cutting milk out of your diet. My son had all your daughter's symptoms and then some when he was an infant. I started doing research when he was 7 weeks, and by 10 weeks I was off all dairy (and I mean ALL dairy -- everything cooked with butter, milk powder, whey, caesin, etc.), eggs, nuts, and soy. Some moms also go off wheat, but for whatever reason, I never had to. It was hard at first, but it made a BIG difference. My formerly gassy, unhappy baby just calmed right down, started sleeping through the night perfectly (I never "trained" him) -- I can't tell you how worth-it it was. I also lost ALL my pregnancy weight and then some in about a month.
At 3, my son is still allergic to milk protein: it causes chronic congestion (incl. ear and sinus infections), mood problems, hearing problems, all sorts of issues. Without milk, he's an extremely bright, joyous little boy. He takes responsibility for his own diet and asks adults to check ingredient lists before he eats anything.
So, I'm sorry to babble about all my son's random symptoms, etc., but his allergies have taught me that milk protein intolerances are wildly underdiagnosed in young children. A lot of behavioral issues that have parents tearing their hair out can be linked to diet. Ditto speech delays.
What I really recommend is that you get a referral to a pediatric gastroenterologist. He or she can really look at your daughter's symptoms, tell you what foods to avoid, and give you a schedule for reintroducing them. I also really recommend consulting with a naturopathic physician at some point -- that's how I discovered that my son's speech delay and constant tears were linked to trace amounts of dairy in baked goods. A conventional allergist's "prick test" will NOT pick this up; please be forewarned.
And -- finally -- please be aware that a milk protein intolerance is different from lactose intolerance. Lactose is a sugar. Milk protein is a ... protein. Lactose intolerance doesn't usually manifest until about age 6, and lactose-reduced products are just as bad for milk-protein-intolerant children as regular milk products are.
Anyway, I hope this helps. Sorry to ramble!
Mira