It depends. You can always call up a police department and just ask if certain things will DQ you. If it will, no sense applying. There are many police departments and from what I have seen, some are more lenient than others. Some only care about honesty and not what you have done, others are more strict.
Now from the point of view from the "felons," let's face it, if they are honest how will they get a job? Lying, I guess, would not hurt them if they are already subject to a DQ. But, why would you want to work for a department that has these kind of standards? Just apply elsewhere.