If the gun is stored in a case, container or safe, it is stored unloaded and uncocked, hammer down. "At rest" if you will... It seems to me if you leave the action open you invite dust, etc and also put continuous compression on the hammer spring. Would this lead to light primer strikes? I don't care to find that out...
A rifle "go" bag will also have an unloaded gun, safety off but the mags will be full. Load mag, cycle bolt, ready-to-go!
Any pistol in a nightstand, desk, toolbox, waistband, briefcase, etc. is intended to be used immediately and is therefore fully loaded with one in the chamber, safety on (if equipped).
PS: I don't currently have a shotgun as a HD gun, but if I did it would be in
condition 1 (fully loaded, 1 in chamber, safety on). I do have a PG in my trunk with a box/5 00buck that would have to be loaded in the tube. The aforementioned go bag is also there...