So basically usage of Bag in AoD is to eliminate the AP requirement of removing and equiping stuff to your hand(s) in inventory?
Yes.
If you equip something directly from inventory, the cost is weapon de-equip cost + item equip cost. So if you equip a bag item instead, you only pay the weapon de-equip cost. If the weapon can be placed in the bag slot the other item came from, they just switch places and you pay nothing.
Same concepts apply when you're switching back again. I think dragging a large weapon over an equipped item will put the item in the inventory per default and not back in the bag, so put the item back in the bag first to save as many AP as possible. Doesn't apply when the weapon is in the bag of course.
At least I'm pretty sure that's how it works...
Dunno how DR does it though, haven't gotten around to it yet. I assume you can just use the item directly from the bag in that? I.e. throw nets without fiddling with (de)equipping and such?