Although most of the time dairy intolerance presents as diarrhea, sometimes it presents as constipation. So, yes, that could be a problem. Most of the time, kiddos grow out of it by the time they are 7.
I'm investigating this issue for my 14 month old. And, yes, my dd eats tons of fiber, prunes, sweet potatoes, beans, peas, etc. and so on. She still has hard poops. I'm considering other types of milk, starting with goat. If that doesn't work, I'll try soy or almond, although soy could cause its own issues. For my dd, I need to make sure whatever type of milk she gets is not lowfat, which is why goat milk might be the answer. (They need lots of fat before the age of 2 for brain development.)
I haven't broached the yogurt/cheese issue yet, although I know there are other alternatives (soy) for those, as well. I don't know if soy cheese is any good, or if a kid would eat it. (I don't think I'd want to.) If ice cream is a big treat, though, try Tofutti. It's non-dairy and still yummy!