Short simple answer to questions #1 & #3 is "YES".
Code of Virginia Section 18.2-308,
#2 - Yes, but from a different POV. The Constitution doesn't tell one what they can do. It limits what the government can do. Fine line distinction perhaps, but a most important difference.
#4 - While any personal defense tool that is not restricted by place or design may be OCd (open carried), OCDO by design/intent is limited to the open carry of properly holstered handguns, promoting and defending that right as we go about our normal every day lives.
We prefer to not use the term "weapon" as that implies aggression, whereas we are about the ability to defend. Normally we say gun, handgun, revolver, or pistol.
If you try to find a VA statute permitting OC, you won't. It is the very absence of a law that makes it illegal that leaves the choice up to you. Is this a great state or what!
Keep in touch. There are other laws that apply, but it is not terribly complicated. Look forward to the chance to meet you and maybe take that first OC walk with you.