A felon cannot be the mayor of DC either but there was one recently. So, I never assume an officer is of good moral character (heck, he might be but I do not know that) until I question.
That's fine and dandy, but you were asking if he had been convicted of a felony. A felony conviction would be an absolute prohibition against being a sworn officer in the U.S.
Its an adversarial process .. I have flustered some officers on the stand asking them these types of questions but most just answer w/o being bothered by it; I think they understand the reasons being the questions ... and if not, oh well.
Since they cannot object, if the prosecutor or the judge doesn't curt the testimony off at the knees, they are more or less required to answer unless they make inquiry with the judge. The judges I have seen in traffic court would shut much of that line of questioning down real quick.
And you do get to cross examine the officer. So I always do so vigorously. And while a traffic case is not required to have a DA, some counties still do.
Not anywhere I am aware of in my state.