Yes, the Stormtroopers on the Death Star were intended to be capable of ground assault, or at least to board and take over ships. Their armor and guns are evidence of this. That kind of armor and sidearm combo really shouldn't be necessary for a mere security detail; they were equipped for combat. Soldiers who are trained for combat will often wear their full uniform (including armor, guns, etc.) when they are on active duty, especially in hostile territory, even if they are in their own base and have no reason to believe they will be going to combat any time soon. The Stormtroopers on the Death Star were just soldiers stationed on a mobile military base, prepared to be deployed into combat whenever necessary.
About the names
Stormtrooper comes from the German word, Sturmtruppen, which means "assault trooper." These were WWI soldiers who specialized in infiltration tactics.
Shock Trooper comes from the German word, Stoßtruppen. Shock Troopers were used as heavy infantry in WWI, which is exactly how Shock Troopers are used in the Star Wars Universe.
Judging by the repeated use of nomenclature to match real world examples of German soldiers, it seems obvious to conclude that anyone identified as a Storm Trooper in Star Wars likely was trained, equipped, and prepared to perform the duties that we would normally associate with a Stormtrooper in the real world. Yes, anyone called a Stormtrooper should be trained for ground assault.