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First answer is no. Second answer is yes and no. There is a difference between hopper/revealer and site-remover. Hopper and Revealer only work when drawn whereas the Site-remover works when it is placed in your Land of Bondage so the ability would activate if you search for it.
Most cards that search for a lost soul is an opponent's card. If they "draw" a lost soul from your deck, then it wouldn't make since for them to "draw" another card.
Each time you draw a Lost Soul at any time in the game, it is immediately placed in your Land of Bondage and replaced by drawing another card.
A search card can be played to get a Lost Soul from your draw pile. If this is done, you must put the Lost Soul in your Land of Bondage, then shuffle your deck.
I am talking about using search or False peace to search my own draw pile for a LS and then placing it in territory and immediatley drawing a replacement card from my own deck. I have done that before. especially searching for the Color guard LS.