With asthma, dust mite & food allergies, need I use 1) latex or organic wool/cotton mattress with or w/o allergen mattress cover & 2) organic bedding?

Dust mite barrier. Covering your mattress with an impermeable dust mite barrier and washing sheets in hot water weekly as well as removing extra pillows and thick comforters are the most important steps to avoid dust mite allergy reactions.
Doesn't matter. The composition isn't important. If you wash your pillows, synthetic fibers are better, because dust mites can't hang on to the fibers as well. But the main thing is to shield yourself from the allergens with a cover. Get a tightly woven cover, not one made of compressed fibers.
Dust mite covers. For dust mite allergy there are covers that are made to be used on pillows, mattress, boxsprings. Some of these are cheap some are more expensive. Google mission:allergy for some good covers.
Dustmite proof both. You can use organic materials if this is your preference. Keep in mind that the dust mites will live better in natural materials such as cotton and wool. You must use a dust mite proof encasing for the mattress, box spring and pillow. If the cover is sealed with solid surface of latex on the inner surface, you need to be careful the expose fabric is not too thick that it creates a home for them.
Mattress cover. Great question. A cheaper choice may be just using the mattress cover.