Apply corn gluten meal (available at feed stores like Agway, it'll be cheaper there than at nurseries) can be applied to the lawn when the forsythia blooms and every few weeks thereafter; it is a safe (it's animal feed, after all) pre-emergent (it prevents weed seeds from sprouting). It does nothing for weeds that are already there, though.
Apply a thin layer of finished compost over the lawn (the rain will wash it down, and the grass will soon grow up to cover it. This is the best fertilizer for your lawn!
Keep your lawn mower on the highest setting, ideally about 3". This may seem long if you're used to putting-green lawns, but long grass on top encourages long roots below. The grass will better withstand foot traffic, and the roots will go down deeper where the water is. A few summers ago, we had a drought in my neighborhood, and my yard was the only one that is green, yet I never water! The grass will grow denser too, and crowd out the weeds.
Hand-pulling is the safest way to remove weeds, use a long weed-puller to get underneath the crown of the roots to pop them out. Dandelions have long taproots and will re-sprout wherever the root breaks off, so be careful with them. This can actually be a pleasant way to spend a hot summer afternoon with your family, talking on the grass, sipping lemonade, ...
Love flowers in your grass! Clover is the best flower to plant, it fixes nitrogen, a natural fertilizer, into your soil. Until WWII, a lawn festooned with clover and other flowers was considered the hallmark of beauty, but the pesticide companies could not find a pesticide to kill weeds that did not also kill clover, so they declared clover a weed. Now after 60 years of indoctrination by pesticide companies, we assume that the "perfect" lawn is a clover-free lawn, but in reality, it's not the healthiest! My father used to chastise me when I'd pull out the clover, but I've realized he is right, they are beautiful! As are the violets, when they are in bloom.
Leave the leaves on the lawn. Go over them with a lawn mower in the fall, to chop them into tiny pieces, they'll break down over the winter to provide natural fertilizer for the lawn. This is the way nature does it!
Thank you for your concern about your dog's health! After an organic lawn-care regimen is established, it is less-expensive than chemical treatments.