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Only Russia can't completely turn off its pumps, and only during winter. They'll freeze. Besides that case everyone else can do it or simply slow them down as they've done in the past.

Will they thaw up and be usable in summer, or does the freezing destroy them?

Not destroy exactly, but cause enough damage that it will require extensive repairs. Please note that this what the Russians claimed some years ago as the reason they could not shut down production during a previous disagreement with OPEC so we should take it with a grain of salt.

Saudi Arabia is constantly turning on and off different oil wells to manipulate the price of oil. This current crash is in large part due to them turning on all of their pumps.

Usually, one can, with no ill-effects. Many operators run their marginally economic wells for only a few hours per day already.