We now have constitutional carry in FL that just went into effect so no permit needed for most places, but having a permit allows you to carry in several places you can’t without the permit. To answer your question, in FL if you have a permit it shows up when they run your tag or DL.
Im my experiences dealing with cops while carrying, some will disarm you, some will not, most have been cool about it, younger cops are more jumpy than veteran cops, give a cop a attitude expect one back is my experience.