As Sloniksp pointed out, once encircled, 6th Army is doomed. You have to maintain an open route and not allow a complete encirclement. There are very, very few cases on the Eastern Front of large German pockets being able to fight their way out successfully. The few that do manage were always ones with a high degree of mechanization and mobility to begin with. 6th Army lacks this.
The Luftwaffe is not going to be able to stop the Soviets or prevent an encirclement. In my version they are there only to provide a means of slowing the Soviets and making their operations more difficult. The ground forces have to hold the pocket open and allow units to escape.

Read the original question again, and ask yourself would the German forces be in far better position to breakout if Hitler had ordered them to breakout on the 25th November 1942 or to wait until Hitler may have ordered a breakout on the 2nd February 1943 (when they surrendered), I know one thing, they may just have pulled it off, or not but they never attemped it, and i bet 250,000 to 350,000 trapped Germans and Roumanians fighting like hell to escaped would be very hard to stop, desperate times call for desperate measure, and desperate men can achieve great many things.

They indeed never tried, but had they the likliness of success seems to have been slim:
1) Although Hitler forbid the 6th army to simultaneosly break out, Mannsteins troops mounted a serious attempt to break the Soviet encirclement. Although his troops were in much better shape, relatively fresh, and well supplied in comparison with the battleworn 6th army, they hardly got halfway, so even with an all out attempt at breaktrough by the 6th army it seems unlikely that more than a few small units would have gotten through.
2) Even when mistaken by such assumption, would it have been wise to break out in November '42? Almost certainly not as the 6th army would have had to leave all heavy material behind and no defensive positions in the hinterland whatsoever had been prepared. Such a breakthrough/retreat might easily have turned into a rout if the Russians had choosen to go in pursuit, (for which they certainly had the means), thus doing more harm to the Southeastern front as already was the case.
3) More important: German panzer troops were still in the Caucasus and a hasty retreat of the 6th army and relieving troops after a successful breakout would have badly jeopardized their position, in all likeliness resulting in them being cut of. 6th army remained at Stalingrad and thus made possible an orderly retreat from the Caucasus.
4) Concluding; Although by no means wanting to bestow on Hitler the title of great stategist, his order to stay put at Stalingrad seems right as anything else would have led to greater damage.

An armed mob, pretty much regardless of size, is going to lose to an organized army. By 2 February there was little left of 6th Army that could be called organized. Battalion fronts hundreds of yards long were being held by men in ones and twos manning individual positions. There was no higher organization in many places. There was no system of communications, no chain of command.
History shows that hastily assembled militia and disorganized regular soldiers lose virtually every battle they fight in. Numbers alone in war mean little. Organization and leadership is everything.

Very good first post, well thought out, intelligently put, worth a reputation point on first time.

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No Rest for the Wicked!

This is a hard one, but when would be the best time to breakout after encirclement, and what method to breakout.

The breakout would have to start almost the next day after encirclement, if that was possible.

For reasons of strength of the enemy encirclement, for sure it seems that any attempt on a breakout should start as soon as possible after the encirclement has been accomplished, as it is evident that the enemy will broaden and strengthen his encirclement forces with every day. This was certainly the case at Stalingrad as the Soviet operation was well planned and well executed.
However starting a breakout at an early moment is not as opportune as it might seem for several reasons. Firstly the encircled troops have to disengage from the front they are fighting on. Secondly they have to change their direction of operation by 180 degrees and have to redeploy for the new task, something that costs time when forces as massive as the German 6th army are involved.
But most important of all: When attempting to break through an encirclement in a retreating way, fall back positions that can be held have to be prepared, and as few in the German command structure had expected a Soviet encirclement operation of this size, (air general von Richthofen seem to have been one of the very few who had pointed out the danger to the exposed and relatively poorly manned Don flank), nothing such was in existance.
The German troops facing the outer ring of encirclement in the West had already enough problems of their own and had to largely depend on improvisations. A sudden influx of massive numbers of worn out, supply less and heavy armor less troops of the sixth army would with certaintly have resulted in utter chaos, with the supply situation rapidly collapsing.
Given the great strength of the Soviet forces in the theatre of operations this would have opened the way for them to put the Germans to rout, which would have almost certainly have also led to the loss of Hoth"s panzer army still in the Caucasus.

Great point. Had the Sixth Army broken out, the Germans in the Caucasus would have been lost. So I guess Hitler did the right thing.

My turn now

What if v.Manstein's relief operation had succeeded after all and made contact, and a substantial part of the troops trapped managed to escape thtough the open corridor?

What if v.M. managed to establish a suitable defensive position, reinforced by the extra troops?

Bear in mind the equipment (tanks, arty, vehicles, etc) that would have to be left behind anyway for lack of fuel and horses.

Not only any number of rank-and-file Ländser would be welcome, but equally as good would the fact that the all-important staff personnel and army specialists would be saved.

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No Rest for the Wicked!

Apart from that in reality Manstein was still very far from his goal when he was stopped, a very interesting preposition!
The first thing that it would have brought for sure is a tremendous boost to German morale. Churchill was right that wars are not won by successful evacuations, but such events have always given new confidence. On the other hand it would have been bad for Soviet confidence and it might be that heads of generals had rolled in consequence.
But from the military point of view, the advantage for the Germans would not have been as great as it first seems.
The number of encircled troops in Stalingrad was truly impressive, but a successful relief operation by Manstein, would by no means have meant that all of them, or even the better part of them would have once more been available for battle. As the corridor to the encircled troops would only have been defendable for a very limited time, not only would all heavy material have to be abandoned, but also the numerous sick and wounded, (even those only slightly wounded, as this would have turned into a forced march by foot escape). Apart from that, the red army would not have taken such an event lightly and would have fiercely attacked and shelled the escaping collumns leading to plenty more casualties.
Second comes that instead of making an orderly retreat with most of their heavy material which was enabled by the prolonged resistance of the 6th army in its surrounded position, the panzerarmee of Hoth in the South would have to make a run for it, losing much valuable material.
Above all that comes that Mansteins position after relieving the Stalingrad troops would have become less then enviable. What makes mass encirclements like the one at Stalingrad such foremost military feats is the fact that the attacking party needs massive amounts of troops and materials as it has to contain the encircled enemy on all four sides. Now, even if the Germans had arranged for a well constructed defense line for Manstein to fall back on things soon would have gotten pretty hot there. The tremendous shortening (and shift westwards) of the front caused by the evacuation of Stalingrad, would have been more to the advantage of the pursuers than of the attackers as they could concentrate all their might on the shortened frontline. On the other hand the Germans would have to do with what was at hand anyway: Given the mauled status of the 6th army few of its units would be in condition for serving as direct reinforcement of the new, lets call it "Manstein line", while the original troops would have pretty much suffered in their relieve offensive.

However, like I said at the beginning, it would have been tremendously interesting to see what would have developed in case this scenario had indeed become reality!

One must not forget that there´s always the danger that the Red Army would let the panzers of the relief force in but not out. Then what would Manstein fight the Red Army with? Or stop the pincers clutching around the relief force if the Red Army attacked them?
Besides with 75,000 men and 500 tanks the German force was not really a force that would crush through the Soviet lines and hold it, I think.

Operation Winterstorm

The SS panzer troops from France would not be available until Feb 1943.

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Considering huge gap in strength, supply, and above all, mobility between the Russians and the Germans, it would have ended up with the total destruction of the Sixth Army, German Army Group A in the Caucasus, and the Forth Panzer army.