Hi ksomom,
Oh my goodness, I am having a heck of a time finding and choosing a mattress!
The first place to start your research is the
mattress shopping tutorial here
which has all the basic information, steps, and guidelines that can help you make the best possible choice ... and know how and why to avoid the worst ones. The first step would be to decide on your budget range and decide on the types of materials and the types of mattresses you are interested in trying (or buying).
are any other brands generally more preferred?
Outside of PPP (Posture and alignment, Pressure relief, and Personal preferences) a mattress is only as good as its construction and the materials inside it regardless of the name of the manufacturer on the label. Having said that ... the tutorial post includes links to many of the better online options I'm aware of (in the optional online step). Some local testing may be helpful to decide on the types of materials you prefer.
And, how about dust mites? My daughter (and possibly my husband) has a dust mite allergy, so it would be helpful to have a mattress that is less hospitable to the little critters. Of course, I realize that we can use a dust mite resistant encasement/cover (which we do currently use on all the mattresses in our home). But it can't hurt if the mattress itself is also a deterrent. 
There is more detailed information about dust mites and allergies in
post #2 here
.
We are very, very, chemically sensitive with big headaches and other issues with strong smells.
Post #2 here
and the posts it links to has more information about safe, natural, organic, "chemical free", and green mattresses and mattress materials that can help you sort through some of the marketing information and terminology that you will encounter in the industry and can help you differentiate between them and decide on the types of materials you are most comfortable having in your mattress and help you answer "how safe is safe enough for me". These types of issues are complex and are generally specific to each person and their individual sensitivities, circumstances, and specific criteria.
Also... which mattress material is the least likely to have issues with mold? We live in Hawaii and it can be a bit humid. I realize that the foundation or surface the mattress is on can make a big difference here, but if each mattress material was placed on the same surface, would any one of them be less prone to developing mold?
I haven't seen any specific information that compares different types of mattress support cores based on their likelihood of developing mold or mildew but if I had to guess I would say wool would be the least likely (although wool mattresses are not particularly common) followed by an innerspring support core (because of the high airflow) followed by latex followed by polyfoam although in most cases with a suitable foundation that has some airflow under the mattress mold and mildew wouldn't likely be an issue with any of them.
The first step would be to decide on the materials you prefer so you can narrow down your choices. There is also more about the most important parts of the "value" of a mattress purchase in
post #13 here
that can help you make more meaningful comparisons between mattresses based on suitability, durability, and all the other parts of your personal value equation that are most important to you.
Also, I should point out that the Tuft & Needle that we are looking into is the 5" one. And this is for our 7 year old, nearly 50lb. daughter. We want something that is cozy (so not super firm) but supportive. I'm thinking a medium-firm would be ideal. Not sure which of the ones we are considering, if any, would meet that criteria. Not that I'd like to add more options to consider, but I am open to other beds to consider if you have any suggestions you think would be suitable. Unfortunately, we can't try the beds out because we are limited on time, and the availability of mattresses in Hawaii is rather dismal.
Post #2 here
includes a number of links to some of the better forum posts and topics about mattresses and children which include some guidelines for children and more information about many of the better options as well but the Tuft & Needle Five would certainly be well worth considering.
Phoenix