At the beginning of WWII, Romania was independent, but then the Ribbentrop-Molotov pact came and parts of the country were invaded by Russians. This pact triggered Romania to join the war on the German side, but there is something interesting when looking at the dates. As part of the Axis, Romania joins the "Eastern Front War" on June 22, 1941, and regained Bessarabia on July 1st, 1941. Basically, Romania could have stopped there and kept the troops around the lost territories (including Transilvania), but instead, Antonescu continued to send lots of troops to Stalingrad for manslaughter. This raises the question: was Antonescu following Hitler's orders? Or was he just in a spree to get more territories?
There was no room for "partial war" in WW2, especially in the East
First, let's make a little digression. There is a certain interpretation of WW2 events, which claims that Finland did exactly the thing you expected from Romania - i.e. (re)captured certain territories it claimed to be theirs, and then refused to advance any further into the Soviet Union, despite German protest. But this interpretation is false, even usually pro-Finnish sources on Wikipedia confirm that :
During the summer and autumn of 1941 the Finnish Army was on the offensive, retaking the territories lost in the Winter War. The Finnish army also advanced further, especially in the direction of Lake Onega, (east from Lake Ladoga), closing the blockade of the city of Leningrad from the north, and occupying Eastern Karelia, which had never been a part of Finland before. This caused Great Britain to declare war on Finland on 6 December. The German and Finnish troops in Northern Finland were less successful, failing to take the Russian port city of Murmansk during Operation Silver Fox.
And after the war, Finland lost even more territory then it has lost in 1940. No matter how do you put it, it was clear to Finns (Romanians, Hungarians, Italians, Slovaks, Croats etc ...) that if Germany loses war, they will lose too. There was simply no chance any of these countries would not suffer consequences of German defeat in such brutal conflict.
Now let's return to Romania. First of all, Romanian armies in the summer of 1941 faced Soviet Southern Front. Due to combination of various factors (better equipment, weaker opponents etc ...) this front did not suffer so badly as other Soviet fronts in the same period. Territories wanted by Romania (Bessarabia, Northern Bukovina and Hertsa) were (re)captured only in the end of July, beginning of August in Operation München with German support, especially from the air. Russian sources about fighting in the South are much more detailed, they include details about various Soviets counter-attacks with some localized success.
After the conclusion of this operation, Romanian forces still held large parts of the front in the South. These operations and fighting along the coast of Black Sea and in southern Ukraine and relatively unknown in the Western literature (except maybe Siege of Odessa). It is important to notice that there was simply no way that Romania could end its participation in war at this moment. Large Soviet forces held in check by Romanians would simply strike at German right flank in Ukraine, thus jeopardizing whole operation Barbarossa.
In fact, events in 1942 that you mention only emphasize this point. Romanian forces in 1942 were not in Stalingrad. Romanian Third Army was more then 100 km away from Stalingrad when it was attacked on November 19th, 1942. The reason they were there in the first place was sheer length of the front. Germans simply could not hold everything by themselves and attack at Stalingrad at the same time. In fact, even with Romanian (Hungarian, Italian , Croat etc ..) support , German front crumbled as it did historically.
To sum things up, when Romania sided with Germany, they did understand they would have to go all the way or suffer consequences of defeat. WW2, especially on the East, was total war, war of annihilation, and there was no place for lukewarm combatants there.
The root cause of Romania's accommodating attitude towards Nazi Germany was a territorial dispute with Hungary. This dispute arose out of the Treaty of Trianon of 1920, which defined the borders of the new Hungarian state, one of the successors of the Austro-Hungarian Empire.
Quite a lot of Hungarians were left outside the borders of the new Hungary. The region under dispute was Northern Transylvania, whose population included many Hungarians, and many Romanians. The dispute had been "settled" by Nazi Germany and Fascist Italy, in the Second Vienna Award of August 1940, when the territory had been removed from Romania and given to Hungary.
The Romanians wanted it back, and while the Axis powers dominated Eastern Europe and the Balkans, the only way to get it was to persuade Hitler that he should change his policy and return it to Romania. Since his decisions were largely made for his own advantage, this may seem a forlorn hope, but Romania really wanted Northern Transylvania back. So policy was to help the Germans with whatever they wanted, and it stayed that way well past the point where it had become clear Germany was going to lose the war and would have no authority over territorial claims.
The policy was eventually changed by King Michael's Coup when Romania switched sides. The Second Vienna Award was reversed after the war, and Romania finally got the territory back.
The Romanians, and Germany's other allies on the Eastern Front, were under German operational command. That's a military necessity to prevent mistakes, clashes between allies and so on. So the Romanian Third Army and Fourth Army were under the command of various German Army Groups.
This wasn't completely one-sided: Petre Dumitrescu was in command of the German Sixth Army as well as his own Third Army for a few months in 1944. The Germans did have to keep the Hungarians and Romanians apart on the Eastern Front; if their units were adjacent, the soldiers would certainly brawl, and there was a risk this would escalate.
The Romanian troops remained under Romanian political command, but the Romanian government's freedom of action was limited. In theory, Romania could have withdrawn from the alliance with Germany. In practice, there were always German troops in Romania to make sure that didn't happen without a fight, as happened when Romania changed sides after the coup.