The 31 chapters of Proverbs, KJV or Amplified(Amplified is very easy to read not so old-timey), even without the aspect of religion the insights gained for business and life in general from Proverbs is profound. 

Fine for religion not so good in business.

Is there anything I need to be careful about other than obviously not discriminating against race / religion / nationality, etc?

But it's not protected under the Federal Fair Housing Act.Here's a link to what's protected under the Federal Fair Housing Act:RaceColorNational OriginReligionSexFamilial StatusDisability   Even apartments.com's own Terms of Service don't seem to prohibit what you posted (again, as long as you didn't reference source of income):Compliance with Fair Housing LawsAll Content is subject to federal fair housing laws, which make it illegal to indicate in any advertisement any preference, limitation, or discrimination because of race, color, religion, sex, physical or mental disability, and/or familial status.

The comedian had a skit about Mormon's. 

There are 7 protected classes at the federal level.Race, color, disability, religion, sex, familial status and national origin.Here in MA, we have 15.  

Even if I find their beliefs crazy as all hell...Do you know if this fear of ghosts is based in a religion?

Questions regarding race, religion etc are illegal. 

Also if this group is looking for a more detailed response  in addition to my adverse action notice as to why we are choosing another group, are the reasons listed above okay to use since they don't fall under any discrimination laws ( age, sex, religion) Thanks for your time and help with this

Age, Religions, Sex, Familiar status etc....Ability to pay you rent is not one of them.As for the debate on whether or not to take a large amount of money all at once that's a whole different animal.