OK, so to gnu, give the dandelions a go if you like, when available. The leaves are definitely decent for salads (if salad is your thing, it isn't everybody's) the flowers are safe but I dunno how they taste unless made into wine, which is alright. I'm sure there's good dandelion-flower wine to be had, but I've only encountered alright.
With hawkweed you are on your own, we await your reports with interest.
Leenia: that sounds right except the getting-rid-of-it bit. Lots of UK gardeners would give up their eye teeth as well as their first-born children, just to be rid of dandelions, maybe they need to move in near you. That gill over ground stuff looks pretty enough but sounds a real pain, luckily you seem to have got its measure.