If your opponent controls a Stranglehold and an effect says "You may search your library . . . If you do, shuffle your library," you can't choose to search, so you won't shuffle.

If your opponent controls a Stranglehold and an effect says "Search your library . . . Then shuffle your library," the search effect fails, but you will still have to shuffle.

Since opponents can't search libraries, they won't be able to find any cards in a library. The effect applies to all opponents and all libraries, including your library. If a spell or ability's effect has other parts that don't depend on searching for or finding cards, they will still work normally.

Effects that tell an opponent to reveal cards from a library or look at cards from the top of a library will still work. Only effects that use the word "search" will fail.

An "extra turn" means a turn created by a spell or ability. It doesn't refer to a turn taken in a sanctioned tournament after the time limit for the round has expired.

If an opponent's extra turn is created while Stranglehold is on the battlefield, but Stranglehold leaves the battlefield before that turn would begin, that opponent takes the extra turn as normal.