IMO, Sidearms, submachine guns and shotguns are awful in PvE.
Sidearms are just a weaker shittier version of the regular auto/burst/scout rifles. The only advantage is a fast switch speed, but that's only really useful in PvP.
SMGs are a shit version of an autorifle with insanely bad recoil. I don't even know what the point of them is. They need to do a lot more damage to make up for the lack of precision bonus damage (not to mention completely missed shots).
Shotguns are just a shit sword.
Are drops based on your current equipment or your total loadout? I know engrams use your best theoretical loadout (even including the vault and alts) but I'm not sure about chests and mob-drops.
I'd like to leave my SMGs in the backpack if possible.
I wish I'd kept my damaged city armour so I could do a quick test.