How would you react to an E2 who "smart mouths" you in formation?

IMMEDIATELY get in their face, lock them into parade rest, and confirm that they are receiving a direct and lawful order. Army regulations (contrary to belief) authorize a reasonable of physical “coaching “ to get him to comply. After that, the PSG, and as a last resort, the 1SG, should be involved. A counseling statement can be used, as well as recommendations for Article 15.

The NCO should be firm, professional, and SWIFT. Otherwise, every troop around will think they can get away with it. Every NCO witnessing the event should band together and put this young turd in his place.

Conjour up a page 11 right then and there

I will mass punish the entire platoon to the pint of feeling dying, but just the platoon, not him, then I make sure everyone understand that every punishment is his fault. I will cancel passes, leaves, and every single time off. I will get them every single detail...at the end, he will feel it either way.

Take sworn statements and push for UCMJ under Article 92. Meanwhile, I'd find as many meaningless, menial tasks for said private to perform until he or she recognized the fact that I am the authority figure here.

After he gets up off the floor,you tell him again! Then put a boot up his sorry ass,that is the way it was done in the army before,

In the "Old Days" it would have been - Fall Out Private and report to the PSG/Plt Ldr's Office, after we fill out your Counselling Statement for Disrespect and Failure to Obey. Of which the E-2 could possibly lose his "free time" (extra training or extra duty) for several weeks, possibly forfeit some money, and quite possibly go back to being a "Slick Sleeve" Private.

From an Army perspective I have to say the information above is not complete enough for me to really judge. My old man was in the service and was all sorts of combat arms in Regular Army and National Guard. My Mother was an Army Reserve LTC running Selective Service with a joint service command. I did Intel. The military is a huge world, and the branch/unit matters...and in this case where it matters is in understanding the soldier in front of you, and those around them. Where I was stationed we almost never saw an E2 (you had to f-up in AIT not to leave as an E3 because of the length of training). Also if you make PV2 dipshit late for work, you are screwing COL OwnsYouBNCDR's ASS by depriving them of 10 minutes of that soldiers time. Further more it was a very cerebral area. Days of PT is not an option. I say this to hopefully break what I read as a chain of very high-school bully oriented knee-jerk reactions.

What is certain is this. The Soldier needs a hard correction. The correction needs to be measured, because if you cannot measure a response then you are unfit for command, or authority. The NCO needs to know his unit well enough to be able to determine if this is a one off/single point of failure scenario in which case you discipline privately...or if it is a culture thing you make the correction to the offending private, but you do it in public where all can see the consequences.

I would also hope the NCO's next level up would take the situation as an occasion to do a pulse check with his unit to see if there are larger problems that needs addressing to include the possibility that the NCO had been conducting themself in such a way as to create the tension.

This one is simple. In front of yours troops, you explain to the dip-shit that this is not the play ground, that you expect him to behave in the same manner as the other troops. And since he does not appear to understand how things are done he can chose between some extra military instruction ( you still have that tool I hope ) which will consist of cleaning a few choice items with a tooth brush, to your satisfaction, during what would have been his normally free time that weekend, or standing tall in front of the person who can issue punishment under the UCMJ.
I say in front of the troops because he smart mouthed you in front of them, and they need to see the results, or they too may start to believe that they can slack off.

Pube patrol

Smoke his ass, if he doesn't care about that kind of stuff, NCOs have authority under UCMJ so if they're not recognizing it, a commander can slap him with extra duty or reduced pay. That kind of disobedience is toxic and has the potential to hurt readiness. find a creative punishment that wastes his free time. If you're his first line leader, unfortunately wasting his time usually means wasting yours. If you're his SL, have his TL supervise some of his corrective training to spread the workload.