Re: adding weeds to compost
I never add perennial weeds that can sprout new shoots from little pieces. This may be OK to do if you are very diligent about making and turning your compost, but I am pretty lazy, and I don’t think my compost ever heats up enought to kill anything. Plus most of these weeds (mugwort, oriental bittersweet, that weedy aster — Symphyotrichum lanceolatum) are pretty persistant.
There are some plants that can make seeds after being pulled up if they have already been in flower long enough. I will have a look through my weed books to see if any of them contain information about this. You would need to look up each of your weeds to see which flowers will be OK.