First issue--- that is a total personal preference. Some people cut them off short so not to see them when they turn brown and others just leave them until they turn brown and they usually can just be pulled right out.
second issue--- you can try putting the prolif in water (with some rooting hormone if you have any) to see if it will grow roots but it may not do it. Now on the ones (the first and second photo) it looks like part of the scape below the prolif is still green. As long as it is still green it is feeding the prolif and I would leave it on there until the scapes shows no green at all.
Third issue---- some critter probably came in and stepped on the fan and broke it (happens here often). I would plant the fan next to the crown because if any of the crown is left on the end of the fan it is possible for it to grow roots and survive (not always though) and the crown should send up more shoots in defense of being broken if it wasn't damaged too badly. There was an older hybridizer (don't know her name) that would ask visitors if they knew how to increase their daylilies and if they said no then she would go step on a daylily and break the fans. She said it would send up 4-5 fans from being broke. I have actually seen this happen when something has trampled through the beds or I accidently have broken a fan off.
