With going down the stairs and off the bed, it was at about this age that I started thinking of it as well. However, it's not until very recently - like the past week or so - that DS has started to understand that it's the best and easiest way to do it. He's 13.5 months now. I would say just keep working on it and be patient with him as he will slowly learn how. I would also stop him every time he tries to jump off and help him get down safely, just as a way of teaching him it's not the right way. I wouldn't expect him to understand that for a long time, but if you are persistent with keeping at showing him then he will one day start to understand.
I didn't really leave LO alone very often at that age to play on his own, mainly he was clingy and didn't want me out of his sight. But, when I did, I made sure there were no electricals - cords, plugs, etc) in sight, no tables with sharp edges and so on. One thing he liked was if I blocked him in a circle with all of his toys and a chair on one side - he could use the chair to pull himself up on, but each direction he went into he found a new toy.
But, I found he was happiest if I took him around with me - he liked to help fold laundry, put the plastic dishes in the cupboard and so on. I also kept secret toys out that he ONLY got when I was cooking and needed him to be entertained. He would be excited to see them because they weren't his regular toys - and that still works at this age!
We have ceramic tile floors in our kitchen now and I worried a lot about DS falling but he's been pretty good - doesn't seem to hurt him any more than the carpet
HTH?