This may be of interest to you. It goes to a book containing two short primary accounts by Joshua Lawrence Chamberlain, of Gettysburg fame, about his experiences at the Battle of Fredericksburg in December 1862. (Note: Link goes to the first account. The second is immediately after.) The first account deals with the night immediately following the fighting, while the second takes place a couple of days later, describing the night of the Union withdrawal.

To summarize the first account: Chamberlain's unit is part of the Union reserve, and is sent forward as part of the last attack of the day. They're the front line when night falls. After dark, Chamberlain delegates command of his unit to a junior, then goes about tending to the nearby wounded with the help of a staff officer. When medical support finally comes, they return to their unit. The majority of the men are sleeping, with some posted as sentries. Chamberlain, seeing that things are quieting down and most movement on the field has stopped, pulls two dead men together as a mattress(!), and using a third as a pillow (!!), goes to sleep. They spend the next day and night there, using the bodies as makeshift barricades, before they're relieved.

So, to your questions. In this case, some of the soldiers did sleep on the field, even to the point of sleeping among and on the bodies of their deceased fellows, while maintaining sentries to watch for enemy attacks. The majority of the army, however, was further to the rear, but they're not back at the camp, which is across the river. (The very start of the second account briefly describes the bivouac Chamberlain's regiment had during the day of the 15th and is probably representative of what these troops had.)

Your remark about the cold is well founded, since Chamberlain remarks that he misses his greatcoat (it was on his horse, which he couldn't take forward), but his men, who were on foot, have their coats and blankets. Said coats and blankets may not have been adequate, given that Chamberlain complains that people were scavenging the dead for warm clothes, and kept waking him up. Given that the average overnight temperature for Fredericksburg on the 13th of December is cold, but still above freezing, it would have been an unpleasant night for the Colonel (at least until he makes his macabre bed), but not really dangerous. Also, it should be noted, Fredericksburg is unusual for how late in the year it was for a major battle. Generally the majority of the army went into camp over the winter, then returned to fighting in the spring.

Now, from the second account we can see what things are like a bit later, after the fighting has stopped but with the two armies still in close contact (Really close. Chamberlain describes how he gets turned around in the dark and winds up in the Confederate line). At dusk Chamberlain's brigade goes forward to relieve another unit facing the Confederates and digs in. Meanwhile (and no one has told Chamberlain this), the Army is preparing to withdraw across the river. This situation is probably closer to what you're thinking of with the camp situation, since the majority of the Union army is either back across the river in their camp, or preparing to withdraw there. Even here, though, they're keeping a force on the line to maintain their gains, such as they are, and to stop, or at least slow, a Confederate attack, until everything is ready for the retreat.

Summing up, at least some portion of the army was kept overnight at the line of contact with the enemy. When things are better organized they might be rotated in and out so they everyone can get some rest. When the majority of the army has just gotten shot up and is busy regrouping, they'll be there all night, and maybe even the next.