From my experience you cannot slag constructors. As far as how ammo drops works it’s all weighted based on how low a players ammo reserves are. So if a player with a low pistol ammo reserve open and ammo crate, it will have a much higher chance to drop pistol ammo.
To answer your second question, ammo that get’s picked up by a player is shared between all players in the game - just like cash and eridium - with the exception of grenades and rockets.
One thing about co-op play, however: the game will not prevent you picking ammo up even if all players have the maximum amount of whatever type is available. So, only pick up ammo if you need it or one of your co-op partners says “grab some ammo for me!”
Note that this is different to BL1, where ammo pickup only went to the person collecting it.
Which is why I cautioned about stopping once you have enough - because otherwise all that ammo is gone and wasted!