Well I think you're right, we usually do dream almost every night. I only had the feeling of those non-dream-nights when I did meditate lots over a weekend for example. This way all that stuff in my brain went by during meditation where the brain is a a very similar state like while sleeping but with your mind awake. It all passes by finally and you can come to the point where it all stops. This of course needs time of course but I had that experience and I guess in that state of mind you have dreamless nights.
About walks I think it's an important point to NOT think about it all while walking, instead let it all pass by and just concentrate on the walk itself, then it can be like meditation yes.
I also had similar problems to you for some time but it just got better and better...and I'm pretty sure it got better because I finally changed my way of live in lots of ways...
You have to find the roots of that problem not fight the symptoms. In my case the problem simply was that I completly had lost the connection to nature and was living "online" more than I really lived. I also think food is an important point, and maybe the most important thing is to find back to calmness and not always hear/see something...find back to the root of all things.

giant