Say, an enemy was shot and killed, and his comrades began to retreat. As the soldiers began pushing the enemy back, a soldier sees a gun, laying in the dirt.
Would he pick it up? Would he trust the enemy weaponry's design? Would he be encouraged or discouraged by his superiors to use it?
Completly yes in several contexts.
In the rush of an extended battle (several weeks like Stalingrad or Normandy) using enemy weaponry was common but it make sense only if the provided guns were in bad shape or out of amunition. Using the enemy's weapons means that it was needed to get compatible amunitions or spare parts to keep it in good order.
The contexts of using the ordnance of a vanguished army. The german high command decided to recycle tanks and guns when they were available in significant numbers as long as spare parts were available. That why Skoda vehicles were used by axis forces and that french captured B1, H35 and Somua tanks were either given to germany's allies or used as training vehicles and used by occupying troops. It was useless to keep heavy ordnance behind when they were needed on the front, so it made sense to use what could be used to keep the german army running at top condition.
On the battlefield as well, the Eastern front gave historians pictures with german marked T-34s and russian marked panzer IVs and soldiers using german mp44s.
During the allied invasion of german occupied Europe, the US supplies make it less common to rely upon enemy captured ordnance but the French FFIs (kind of a militia latter more militarized) exterminated in the Vercor or Polish fighters at Varsaw used a wide variety of small arms including germans.
In 1944-45 France, the American, English, Canadian and French divisions could not spare too many troops and weapons to besiege the atlantic pockets, german strongholds like Dunkirk, Lorient or Saint Nazaire that german command had turned into submarins bases, the task was given to the then more organized FFI that used what the US could spare, taken back pre-1940 french weapons and captured german tanks. I would like to give you the links of the german T34s and the french FFI panther I will look up in a book once at home.