Did they bury Civil War soldiers?

Yes. Most Civil War soldiers were burried in mass graves. These mass graves were simply hastily dug trenches, maybe 4-5 feet deep at the maximum. The dead were then cast into these trenches and covered with soil. Mass graves were typically left unmarked, or if they were marked, very little information was left, basically leaving the dead soldiers as 'anonymous'.

Some soldiers had the priveledge of being given a proper burial, but they were in the minority. After a major battle resulting in thousands of deaths, the main objective was not to mark who exactly had died, but to simply dispose of the bodies and move on.