Need a Little Help

I'd say YES on all three. 
On the third question, I'd say YES but I would recommend gun owners be responsible and image cognizant, like not leaving a loaded gun in plain sight on the seat,or on a gun rack in the window etc.   No "In Your Face Cause I'm Legal and The 2nd amendment makes it legal so SCREW YOU - I'll Do What I Want!"

yes on all 3.  If your doing nothing wrong, then the other guy/gal should keep their mouth shut.  Tell the guy from sweden... if he don't like our ways, then get the hell out of MY country, take his wimpy arse and go home.

I believe you have a moral obligation for all 3 questions.  You have decided to carry a weapon and have your permit to carry.  Once you have taken this step, things are different.
On the third question if you have a legal right would depend on the state.

Here is some info on Sweden gun ownership from Chuck Hawk.  Tell that Swedish guy, the Nazis were against gun ownership and look what happen to the 5 million Jews they murdered.  The majority of Europeans don't know what freedom is, they live in a form of socialism were the government controls many aspects of their lives.

http://www.chuckhawks.com/hunting_sweden.htm

What gives him the leagle right to know whats in my car?

This may sound extreme, but I believe that if you are a law abiding citizen that is skilled in the handling of a firearm, you have a moral responsibility to not only protect yourself and your family, but those around us.
As far as your discussion, all of the states that enacted right to carry laws immediately saw a decrease in violent crime. This info is attainable through I think the FBI or you can try Google, at any rate it's fact so if you can find it that would end your discussion.

I passed your question on to a buddy of mine who knows a lot about guns and the law.

I hope it helps.
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First off, the Second Amendment doesn't give you anything.  Then, state laws are subordinate to that charter, the Constitution of the United States.  2A recognizes a limit on government (...shall not be infringed.)
 
A moral right?  What does that mean?  Does our Creator give each of us humans life and liberty? Yes.  Go as far back in man's word as you wish, maybe to the Magna Carta, or just to the Declaration of Independance, and you can see that (wise) man has always recognized that, at least on some level.  The DoI recognizes "that all men are created equal, and that each of us is endowed with certain unalienable Rights..."  Again, our founding documents grant us nothing; that comes from our Creator. So, if you were granted life and liberty on this earth and you choose not to protect it, fine.  But, that doesn't mean you can choose to forbid someone else his right.  That would infringe upon his liberty.
 
Moral right to own a gun?  Again, what does this mean?  Are we asking if we have a right to property? Then, yes.  Is it moral to own property? Yes.  Pedo-porn? No.  That infringes upon the unalienable rights of the child.  Slave-made guns? No.  Guns? Yes.  Guns for bad intent? No.  If what you are doing infringes upon the rights of others, you don't have a right to do it. 
 
A loaded gun in a public parking lot, regardless of what the owner wants? Well, if the lot is public, YOU are the owner.  If the lot belongs to ABC retail and they say no guns, fine; the property is THEIRS, and they have the right to their property.  Same as you can say no swearing in your house, or no political talk in it.  If you take that gun onto a parking lot that says no guns, the owner has the right to have you removed.  However, there can be no prosecution under law, because the rule isn't law, it is a private policy. If the parking lot is for something private, especially behind a gate, or with no trespassing signs, or in the middle of some farmer's field, for that matter, and you go there, you are trespassing and the law accounts for that, because the farmer has a right to his property (without you on it).
 
Moral or legal right to do so?  Those are two different questions.  Moral: We are granted life and liberty by our creator.  With that comes a right to protect it.  After all, if we can't protect what we have, we don't really have it (or at least for very long).    Legal: Legality is relative.  Each jurisdiction has different laws.  Heck, in at least one town (Kennesaw, GA), it is illegal to NOT own a gun.  If that doesn't prove the relativity of legality, I don't know what does.
 
To say 2A is outdated is silly.  The right to life is outdated?  The right to express yourself is outdated?  The right to a speedy trial is outdated?  Next thing, the rights of states will be superceded by Federal laws.  Oops...  If you don't have the right to keep and bear arms, you don't even have a right to your own life.  And that is a sad situation.
 
Lastly, parking a car with a gun in it may be foolish, since someone else could get to the gun.  Keep the gun on your person.  Not in a bag that you could forget, but on your body.