When an E-4 calls me "hey" or "hey man" knowing that I am an officer, should I bother to correct him?

Absolutely, even after separation we as Soldiers are still bound to Our Oath, Sir

If you are outside of work. You are a civilian. There is nothing wrong, with addressing you as a civilian. Now when you are both in uniform and at work. It is mandatory for the SPC to salute you and address you as Sir. You earned that right. That is a sign of respect.

O-1 to O-4 I can create a good relationship with if they're willing. If they demand customs 24/7 then I can't say that I'd be privy to them. Can't respect the person after that, only the rank, and only if you can find me again, because I would make it out of my way to never speak to or see that officer ever again.
O-5/O-6 can be casual with me, but I would still stand professionally and speak to them professionally. Some casual language may slip between the cracks. Same rule about them demanding C/C as above.
Generals are balls to the walls customs and courtesies 24/7, but since I never see or interact with them, I can't really say. I've spoken to my CG once while on DIV staff duty casually one time, as he just wanted to small talk with us. Really liked him after that!
It depends on the relationship you want to make with your soldiers. Do you want to demand C/C and seem like an a** and the similar minded ENL members won't be privy, or build a relationship with your soldiers and really be a leader that your soldiers can be comfortable talking to and trusting. Obvious disrespect should be corrected, not the soldier showing you that they like you/respect you. If I say to my 1LT "Hey sir, how was your weekend" he tells me, and I respond "yeah man, I was up awfully late last night working on my project", that's obviously NOT disrespectful. I would be showing him that I am comfortable talking to him. If he demanded standing at attention and calling him sir, I would only ever go to him if I NEEDED to. I would never want to small talk, or speak to him about my personal life, ever! As I said, ots whatever relationship you want to have, really.

Is this a serious question? Unless you are Guard or Reserve, and even then it should not happen IMO, then you are undermining your own authority and, frankly, the authority of every O, WO, and NCO in the eyes of this punk. You don't have to be an obnoxious ass but you can and you are OBLIGATED to correct this troop. Furthermore, offense intended, WHY THE F**K DO YOU EVEN THINK THIS IS AN ISSUE THAT NEEDS TO BE ADDRESSED HERE?!!! WHAT KIND OF OFFICER AND LEADER ARE YOU TO LET THIS SLIDE AND TO NEED THE CONSENSUS APPROVAL OF OTHERS? Frankly, the greater problem is you. Man up and be a leader that demands and earns respect, first by your office and then by your personal merit.

Should You Correct Him?
Holy Shit, That Shouldn't Even BE A Question....BUT, Since You Asked. ..
"OH HELL YES"
Immediately After Kicking His Ass.
Providing You Were In Uniform.
NO Other Reason Is Acceptable.. ZERO

USAF 1961 - 1965

Uhhhh, ? If you really don't know the answer to that, you should immediately resign your commision.

Correct in private, praise in public.

I would tell him that when we are both off duty it is a common courtesy to call me Mr. R, NOT "hey" or "hey man". I am just going with what I was taught growing up as a Navy BRAT and the interactions my E7 father had with officers that were not always in his direct chain of command.
As a member of the U.S. Naval Sea Cadet Corps, I was again told that when people were off duty it was fine to refer to them in civilian terms.

I have always thought of the LT as the mother of the platoon. Daddy was always an enlisted member and should be doing the discipline. You should probably talk with your NCO's and let them know there is a discipline standard and to make sure the soldiers are adhering to it. Basic customs and courtesies are a bed rock for disciplined soldiers. Let the NCO's handle it, but correct him with a Sir, if it comes out of his mouth. If you dont they wont see you as a superior.

Hello YES