Anyone know what Army Regulation (or any branch) that allows a higher-ranking officer to call a lower-ranking officers by their first name?

There is no regulation that allows or approved of this. People should definitely be weary of fraternization but i think that's officer to enlisted, not officer to officer. Although rank shouldn't matter honestly.

Anyone can see your rank. If a higher ranking Officer or in my case NCO calls you by your name, it has the effect of: "Your name is known to us. I have heard tales of your deeds, my people sing songs of your glorious victors." Or something like that.

Some units work fine this way. Special ops units tend to do this quite a bit especially on missions where the ranks are kept quiet. Also in those units, the most qualified to lead a particular mission gets put in charge, not always the person with the highest rank. The USAF & National Guard also does this. In some units it does not hurt things, other units, it does create a lack of discipline.

This may be depend on your environment. In a medical situation and under stress we might call someone by their first name. This is true in all Combat situations too. How about in private social
situations? I say OK. There are questions here too. For example. A very Sr. Officer calls his CSGM by his first name and asks a question. The CSGM replies but does not use a first name and uses Sir! (Situation Normal)? There are many other situation like this.

However, I do draw a line here. Referring to others by their first name in the office, field, Command Center, formal briefings, in front of dignities,.....the list goes on. May appear to be very unprofessional
and degrading to some. I once overheard a mid-level NCO (E-7) say to an officer. "Sir, I have a title, I earned it. My name is MSG. John...….."

Formal name, rank or title should be used when in doubt.

This isnot allowed nor professional in public

Don't know of any such regulation and during my time in the Army I never experienced such a situation--but I did experience the reverse: While serving with a joint protocol office in Korea (8th US Army), an Air Force major serving with us once told me to feel free to call him "Charley" in the office (apparently this was not an unusual procedure in some Air Force facilities at that time--though within an office environment only, not outdoors). There was also an Army major in our office, and our boss was an Army LTC. I respectfully told the AF major that, while I appreciated his offer, I would never be able to feel comfortable doing what he suggested (and never would have even considered trying the same with the Army major) and so preferred to call him Sir.

Internally, common names was used however among enlisted, professionalism was priority.
To call a sergeant by his given name destroys respect, when used by officer, among enlisted.

This standard OP Chief Edgar; however, the opposite constitutes an offense in the uniformed services. There is no regulation that stipulates a superior officer should or must address a subordinate officer and non-commissioned officers by their individual first names; I haven't come across such rules yet. It is a fact that personalized contact with subordinates encourages subordinates to perform duties at peak performance. In so saying, the contact must be positive and inclusive rather than negative and exclusive. Do let me know the regulations, once you find them. Thanks and great question.

It exists in at least one branch.
Air Force Instruction 36-2618:
“ rank includes an official abbreviation and term of address, however Airmen senior or equivalent to the member may use first names and/or call signs.”

Even at the Senior NCO rank, first names can and are used. Not as much as officers...Never in front of subordinates. I used first names if a Soldier was of equal rank or 1 rank below me during counseling sessions.
There is no regulatory guidance other than courtesy and common sense.
Officers use the first name frequently. Staff meetings at the Battalion level or Brigade level use first name basis very frequently. Not a big deal. Depends on the situation...but IMO should be used privately...