I unload my lever guns by pushing on the loading gate, which lets a cartridge pop part way out. It's easy to pluck them out, and fewer end up falling into the mud. I'm a fumbler, I know one can easily catch hold of cartridges as they eject, but it's even easier to take them out the port, at least for me.
I pack my levers with one in the chamber, after lowering the hammer to half-cock. I learned how to handle lever guns that came out long before cross bolt and tang safeties, and I don't use those safeties now. They are probably a good idea for someone that's just learning their way around a lever rifle, though.
Usually, I pack them in my hand, rather than slinging them. It's especially easy to carry a Winchester 94 by the receiver, and I've never carried one any other way.