Allergy warning: May contain nuts. check with your attorney and your local regulations.

are they suffering from allergies or is there a bad smell to the place? 

We inform them of our zero policy and to think about if the people that rent behind them might have a pet allergy, but that's all you can really do by law. 

I’m wondering if the dander would transfer into the other unit via ducts and cause potential allergy problems for that tenant.

Bigger pockets:One of my new tenant’s son seems to have allergy and asthma problem.

I use hardwoods they can be easily replaced with sections and there seems to be more people who don't like carpeting due to allergies. 

AND we told them it was okay to keep their cat, even though we were only allowing dogs (cat allergies), so I feel like this is pushing it.If we take chances with a stranger, there is no guarantee that they won't just smoke away from the unit and we won't know if they'll take care of the unit as well as our friends.I say no; husband doesn't care, but will back up whatever I decide.

A test performed by you will likely not inspire confidence by the tenant as it is not an independent third party.Mold out of normal range levels can sometimes by some seen as not that bad and with others that have more sensitive allergies or other existing health issues the effects can be felt much greater.

Think people with allergies, kids, etc.

I want to move to Alaska because I have a rare allergy, its to the heat and sun.