Because the Commonwealth of Virginia has no law prohibiting open carry you do not need to have or to carry a CHP/CHL when openly carrying, including when your handgun is secured in a container. Note that the container, e.g., glove box, center console, etc., does not have to be locked. Having said that, if you are traffic stopped and the LEO knows you have -- and demands that you present -- your CHP/CHL, prudence may dictate that you present it, even if you are not carrying concealed. There will most likely be advice following this post that indicates that you do not have to present it on demand (because you are not carrying concealed) and that you should decline. Some of us who openly carry (and who are not also concealing) will not carry their CHP/CHL with them -- just make sure your openly carried handgun does not become inadvertently concealed by your coat or jacket.
I'm not certain why you call out a separate case for "driving to the range." Open carry doesn't care where you are going -- to the range, to the 7/11, or to the bathroom.
As for your last question, if you are traffic stopped, you are already being detained. For "investigative" questions regarding anything unrelated to the reason for the traffic stop, you'll have to decide if you want to answer directly, not answer, be evasive or change the subject ... but what you may not do is lie. The (somewhat vague) citation that shows false statements to be a misdemeanor is in another thread.