WOW!! This entire thread reeks of paranoia!!
Been fulltime for 9 years and have NEVER been asked that question! However, we are not living on the street...we stay in campgrounds or approved boondocking areas. I see absolutely no reason to lie about my circumstances.
As for searching my RV...the answer would be not without a warrant...simply because of reasons stated above...if something out of the ordinary is found you have no recourse.
As for not trusting LEO's in general...that without a doubt is pure paranoia...when you treat them with the same respect you demand they will give you no problem...at least that has been our experience.