Is using toothpaste with fluoride absolutely necessary? I am allergic to fluoride and use a fluoride-free toothpaste. Is this alright for my children to use as well, or should they stick to using fluoride toothpaste up to a certain age? .

No. Using a toothpaste containing Fluoride is not absolutely necessary, but you need to be more careful about having good oral hygeine and healthy diet low in sticky, high charbohydrate foods and acidy drinks. Soda, fruit rollups and similar products should be avioded. Only about 1% or less of the population has a true Fluoride allergy.
Risk of Decay. Just know that when you don't have fluoride, the risks of decay increase. You can read what the american dental association stance on Fluoride is, plus all the studies they use to back them up.
Xylitol. Introduce your children to xylitol. Use of 6 grams a day has been shown to decrease tooth decay in children. This can be in gum, mints or even candy and is available in some supermarkets, dental offices, and health food stores.
Flouride. Yes- you need to worry about your child swallowing flouride. Checkout the products at www.Carifree.Com for some safe alternatives.
Fluoride . Fluoride toothpaste is not an absolute necessity so long as you brush your teeth well. If you're allergic to fluoride, make sure your children aren't allergic to it as well - go see an allergist. However, Fluoride toothpaste have been shown to strengthen the enamel on teeth and hence reduce cavity in children.