There are great options for prefinished flooring using Green Guard certified finishes from companies like Baltic Wood Floors and there are great low VOC wood finishes from companies like Vermont Natural Coatings.
Your best bet for an easy, cost effective and quick install with a durable finish is probably the prefinished option but I have done both options depending on timing and how custom someone wants to get with colors and sheen.
As for the carpeting, try not to shake and disturb it as much as possible when removing. Roll it up in the space from which it is being removed and try to get it directly our of the house through a window in the same room so it is not dragged through the rest of the house. If you have to take it through the house, I'd suggest tarping your path and temporarily sealing any openings along the way to other rooms to prevent contamination.
For more information:
Read "What is the best way to remove carpeting and treat the concrete beneath it afterward?" a Q&A answered by Erik Henry.
Also, check Kati Curtis's Q&A "Is an engineered hardwood floor that is FloorScore certified a good choice for the bedroom of a severe allergy sufferer?"