If you are lactose intolerant as opposed to milk proteins that actually gives you a limited amount of dairy - butter and some hard cheeses like parmesan. All you have to do is check the nutrition stats for 0 sugars. Be careful with US labels though as I believe they can ‘hide’ carbs if the portion size has only a small amount. Ideally, check a label that tells you what is in 100g.
Eggs - well that is something else entirely.
To confirm both, the best thing to do is an exclusion diet and then test these and other foods to see what reaction you get.
My advice would be to start things simple (you could actually use it as an exclusion diet) for say the first month - fatty cuts of protein, leafy veg and oils and butter. After the month you could then test eggs and hard cheeses. Start with the zero % like parmesan but you could also test up to say 1 or 2% and see what (if any) your tolerance is.
If you have these allergies, there is a chance you have more so you could test these too - nightshades (tomato, aubergine, peppers), nuts (esp peanuts and almonds) and anything else you might be suspicious of. It would be a great way of killing two birds and getting healthy at the same time.
Good luck whatever path you take.