To #1 :
They threw him into the clink.
They tossed him in the clink.
He got himself thrown into the clink (for a reason here).
He got (himself) thrown into the slam.
He was/is imprisoned in the Guantanamo prison/in the Gitmo/GTMO.
He is held at Guantanamo prison/(prison) facility.
To #2 it could (colloquially) be "he was thrown into the clink; at Guantanamo. But se don´t see it usually this way in the mainstream media, since it is usually mentioned when referring to sensitive issues.
One instance is on The Intercept:https://theintercept.com/2019/03/03/guantanamo-bay-carol-rosenberg-intercepted/
Today, under the Trump administration, more than half of the 40 men who remain at Guantánamo are still being held indefinitely without a charge or a trial.
I hope that deals with how to express what you wanted to.