I bought a Lizard Minion trap recently, and I know it has to be in your inventory for it to work. If I lock the item, will it still work?

Yes, being locked just means that no random event or whatsoever can take it away from you. It will still be shown in the auctions, so pay attention if you ever create one.

Yes, I've always had mine locked and had multiple events. I also have the spray. Having both gives a 100% chance you'll get the item every time. Locking both does not change this. :)

yes. Locked items still function like unlocked ones, if they have a passive function. There was a time-travel based special event one Masquerade, and you needed to keep your Device locked in Inventory, or the Rotten Rogue could get it.