If this is a deep draft and the Kings pick 3-7, should they be looking to trade down and go for quantity?
Generally teams only trade down in the NHL Draft if they think they could still land the guy they would have taken anyways plus take an extra asset. Trading down for multiple picks is a sexier option in the NFL Draft since you can reasonably expect a 2nd/3rd round pick to contribute to your team almost immediately. I think NHL teams are a little more realistic than us fans with regards to the likelihood of a 2nd round pick panning out. This is typically why we haven't seen a ton of movement in the top 10 of the NHL Draft.
Just by going by my time following the draft since 1995, the team trading up in the first round has tended to do a lot better when actual names are attached. People tend to remember when the opposite happens (ie, Anaheim trading down and getting Rakell/Gibson) but that seems to be the exception more than the norm. I'm always reminded of Anaheim in 2008 trading #12 for #17 and #28, then #28 for #35 and #39. On paper that seems like good value, but they used the extra picks on Nic Deschamps and Eric O'Dell (who they didn't even sign). Allegedly they thought they could trade down and still get Erik Karlsson at #17 only to get sniped by Ottawa.
In 2008, Garth Snow wanted to restock their barren prospect pool and they figured they could trade down and still get Josh Bailey. After several trades they turned the #5 pick into:
#9: Josh Bailey
#40: Aaron Ness
#74: Jyri Niemi (wasn't signed)
#102: David Ullstrom
future 2nd round pick which was traded (Ducks took Matt Clark with the selection)
A lot of the trade downs end up looking like that where the team just doesn't do a ton with the extra pick(s). Usually the extra picks aren't worth the drama of potentially missing out on your target.