The location sounds good for at least spring

if it’s a very hot summer you may have to provide some shade during the late afternoon early evening hours to keep from having to over water.
Yes, just knock the weeds over and start with the layers of wet cardboard/newspaper. I have landscape timbers around mine to keep the layers in place, but that isn’t absolutely necessary.
For green waste if your grass is still green mow your yard and put that on there. I run short of green waste too, but it all works out just fine.