My children have managed to do most of the things like that you have described in the OP.
My house is often a mess by the end of the day (and I certainly don't do nothing about it), the two year old went through a right spate of drawing on walls, they lose toys and break them sometimes, and yes my two year old surprised me a couple of times managing to open the front door and get out the house, to my complete mortification (there were circumstances).
I'm a single parent, so I have no one to tell me off about these things but myself, which I do. But I think you can be too hard on yourself (and perhaps your DP). Young children do these things, and when something bad happens, you adjust the way you care for them to take it into account, and sometimes maybe it takes a few adjustments to get it right, but it doesn't mean you are bad and don't care for getting it wrong in the first place. I know some people just naturally have it, the common sense and the drive to be really great at parenting, some people are great parents (as in they love them to pieces) without that though (as in they lack natural ability... ahem... me).
Maybe you have to moderate your approach, I was never a natural parent, and someone getting frustrated with my ability would only serve to make me shut down and feel defensive because I was trying my best. Instead I was introduced to several ideas which helped to increase my confidence and helped me get sorted. I am still a trial and error mum, but I like to think I've improved a lot.
Introducing routine really helped to get things on a more even keel for me. If you feel your family is lacking a routine, work together in a non-confrontational way to develop what you have into something that works a bit better for both of you.
I can understand you really wanting him to see it from your perspective and to do it as you would think is correctly, but I think you need to communicate and, yes, work together towards a mutual goal, not just the goal that is in your mind as to how it should be done.