1. Shouldn't think so. Normally ISP's will only kick up a fuss if the file is either being downloaded or uploaded. Generally only uploading (sharing) AFAIK.
2. (a) Go to LW's menu bar, Tools>Options>Filters>Content & enable this function. (b) Check License requirement (click on link)
3. No. Program options are identical. Their support won't help with copyright content. But might advise how to stop your children. But that's probably not too different from what I've already suggested.

Whilst you said you searched their shared folders, instead, search the computer. And search for terms like harry and potter individually or even the titles of those movies. If you're using the LW beta 4.13 then there's a chance they were downloading a torrent file which is shared as it's downlding. Also, because LW allows you to save to or save as ... the file might not be in the shared folder but elsewhere & they might have renamed it something else (in which case you should search the computer for video extensions such as .avi, .mpg, .mpeg, .wmv, .mov, etc. Also check to see if there's any other p2p sharing programs on your computer.