I've come across the "Add to List" option on player card pages and it simply means that the particular card will be added to whatever list you've created. I've not encountered the "Add to List: All Cards" option, though. Where do you see this? Here's the FAQ on Lists.
"Move to Another Collection" is available if you have Multiple Collections other than the default Main Collection setup, and it allows you to move cards to whichever collection you choose.
If you click on "Add Multiples" on any card you'll find two sections. On the bottom section you'll be able to add condition to all dupes of that particular card. I don't believe there is an option for a whole set, though.
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“Aslan didn't tell Pole what would happen. He only told her what to do. That fellow will be the death of us once he's up, I shouldn't wonder. But that doesn't let us off following the signs.” - Puddleglum in The Silver Chair by C. S. Lewis