Cali Bamboo is low VOC, 3x harder than oak and obviously better than carpet. I'd go with a distributor though as I didn't like their direct sales people.

Armstrong Linoleum (linoleum - NOT vinyl) may be an option, at least for some of the floors.

Quoting from the Armstrong website:

"Environmentally Friendly Linoleum

Among the âÂÂgreenestâ of floors, linoleum is made from natural materials: linseed oil, resins, recycled wood flour, cork dust, limestone and mineral pigments, mounted on jute backing. ItâÂÂs naturally anti-bacterial and biodegradable. Sheet linoleum is sold in two-meter (or 6â 7âÂÂ) widths and several thicknesses."

I used it in a 1929 kitchen. Was a very good match for the age of the house. The only odor was a linseed oil smell that dissipated in a day or two after installation.

One precaution: Linoleum takes a lot more care than sheet vinyl. It needs polish applied at intervals to keep it shiny. Armstrong sells the needed maintenance products.

Here is a link that might be useful: Armstrong Linoleum

Any prefinished domestic hardwood would be a suitable choice.

To my knowledge, categorically stating that "engineered hardwood has particle board" is incorrect. Dismissing all engineered wood flooring due to this might be a mistake. I can think of only one manufacturer (Shaw?) that has engineered hardwood floor products with a composite center core. It's more like MDF but I guess that's technically considered particle board. Shaw does sell a lot of private label stuff so you may see multiple brands made with this, but they're all made by Shaw. I can think of one other product that has or used to have particle board fillets at the ends where the T&G profile was milled.
I'm not aware of what makes a product with particle board unsuitable. Typically engineered wood flooring is made of layers of wood cross laminated and held together by glue. There is, or at least has been an issue with formaldehyde in the adhesives that holds the plies together in engineered wood flooring. Some have moved to using glues with no added formaldehyde. This might be something to look into.
The same is true of Bamboo. Bamboo flooring is made of thin strips of Bamboo grass held together with glue. The strand Bamboo, which is harder and heavier, is made of the strands that weren't wide enough to be used making Horizontal of Vertical Bamboo so they bundle it together with an epoxy and cure it under heat and pressure. Here's a link to the Bamboo brand mentioned above.
http://www.calibamboo.com/low-voc-flooring.html

How about porcelain? Can't get much "friendlier" than that....

Tile would be #1 plus its so easy to keep clean..almost sterile. Hardwood is a close second. I'd go with porcelain or granite...or oak.

Thank you, everyone, for all of these thoughtful responses. I have a lot more to work with now!

Porcelain wood plank tile! I just did my entire house.

BigKitty, thanks for the thread. I am watching this discussion. Please do let us know what you decide.

porcelain wood plank tile is intriguing.