In general, unless you are hunting with well-behaved pointers, and the birds decide to play nice, you aren't going to have much time. A round chambered, with the safety on and the muzzle pointed up and away from everyone (and everydog) is most common.
As Steve says, if you fall, it is your responsibility to keep that muzzle in a safe direction. I can't begin to tell you the number of times I've landed on my butt, my side or my back... but the gun has ALWAYS stayed up and in a safe direction.
Assuming you are using the safety, make it habit to check this frequently. In many types of cover, it can get knocked off "SAFE". Again, it is your responsibility to keep your finger off the trigger and protect the trigger (keeping twigs, branches etc... from entering the trigger-guard) at all times.
As far as unloading when crossing obstacles... if I even have a fleeting thought of "I wonder if I should unload first?", then I do it. Better safe and alive than sorry and wounded or dead.
If I'm taking a break and need to lay down my gun, then I'll open the action and lay it down on the ground... not rest it against a tree.
One thing to practice with an unloaded gun... sometimes birds will fly directly at you. Often, it is better to turn and take them going away. When you do this, make sure that your gun is pointed straight up when you pivot. You do not want to sweep any hunting partners as you move around. Make sure to practice this alone with an unloaded gun until it is second nature. Otherwise, pass up the shot.
