Warning! Completely subjective descriptions ahead:
In my experience the best approach to clearing your mind is not to try and actively clear your mind (i.e. thinking about not thinking) it is training yourself to let your thoughts be more transient.
When I try to meditate I try to let all my thoughts slip. Of course, just like you I will hear a noise or some thought tries to worm its way in, but when this happens I try to relax (physically, there is a wonderful link between relaxing your muscles and relaxing your brain. Instead of forcing your brain to relax, trick yourself by relaxing physically) and let the thought go.
Of course, thoughts keep floating into my mind of course, but the longer I keep the meditation up the more transient they become. At some point they even stop being thoughts because they exist to short to even have any content or feeling associated with them.
I have noticed that with practice the time and difficulty of achieving this transient state go down, making it easier to just start. Early on I would need to be in a quiet spot alone, nowadays I can get the process fairly far even with people in the same room talking (if not to loud).
Anyhoo, I hope that description maybe helps you!
TL;DR - You don't stop thinking or actively try to shut down your brain, you passively let your thoughts become more transient.
Yup, that is exact right. The guy who taught me that it was about 'being the stillness' (I know, I know, but this was they 70's). Lets say you are looking at pond, smooth as glass and a pebble falls, ripples spread out. You can't stop the ripples, you can just note them and wait for them to pass. So you note the thoughts as they pop unbidden into your brain, and then wait for them to pass. If you were in a horror movie and the killer was walking in your house, you'd be scared to death trying to be invisible. That is 'being the stillness'. Actively not thinking.
"I struggle with even imagining what it's like to not think about something. Something new or something old. Anything at all. I will note that I also have trouble sleeping for the same reason, ..."
There were lots of people who fell asleep in our class. I did too a couple of times. One of the party tricks you can do while meditating is what some folks call 'lucid dreaming' which is you are balanced on the knife edge between consciousness and unconsciousness, you can learn to let your unconsciousness start up without leaving consciousness completely behind. Me, I end up going to sleep if I try that.
But having a lot of stuff happening at work, at home, and in the world, sometimes I meditate to go to sleep. But its a two step process, meditate, then once there allow yourself into the unconscious state rather than holding on to the conscious state.
There was an engineer at Google who had dug up a lot of interesting references on meditating. Some folks have decided it is a form of self hypnosis since the mental state most closely resembles that of being hypnotized.
>there aren't any big secrets here you are simply training yourself to stop thinking without going to sleep
This is definitely the first hurdle I encounter. The handful of times I have attempted this, I end up thinking about not thinking (isn't that silly?), or a song comes into my head, or I hear a noise in the distance, and attempts to clear out those thoughts lead to, well, more thinking!
I struggle with even imagining what it's like to not think about something. Something new or something old. Anything at all. I will note that I also have trouble sleeping for the same reason, and when someone describes "the feeling just as you fall asleep", for me that's a synonym for stress and frustration. I can never really remember how I eventually fall asleep.