Thrain in Dol Guldur

[Zigûr]

I never even imagined Thráin being "interrogated" especially. I always assumed that Sauron simply arbitrarily tortured him or had him tortured, took the Ring from him, probably in person, and had him thrown in the dungeons to die.

I think we get the impression that Sauron was looking for Thráin, so once he found him the first thing on his mind would presumably have been the Ring:

Little is known of what happened to him afterwards. It would now seem that as soon as he was abroad with few companions he was hunted by the emissaries of Sauron. Wolves pursued him, Orcs waylaid him, evil birds shadowed his path, and the more he strove to go north the more misfortunes opposed him. There came a dark night when he and his companions were wandering in the land beyond Anduin, and they were driven by a black rain to take shelter under the eaves of Mirkwood. In the morning he was gone from the camp, and his companions called him in vain. They searched for him many days, until at last giving up hope they departed and came at length back to Thorin. Only long after was it learned that Thráin had been taken alive and brought to the pits of Dol Guldur. There he was tormented and the Ring taken from him, and them at last he died.

If Thráin was wearing the Ring at the time it would have been a simple matter for Sauron to take it and see no further use for him.

The "torment" may have been the unfortunately arbitrary torture that many tyrants would impose on their victims as a sort of "punishment" for the mere fact of being an enemy. Sauron may not have needed to torture Thráin to force him to give up the Ring, as he could have been searched easily; he may have just tortured him for fun, as seems to have been Sauron's way since the First Age.

I almost think it would be typical of Sauron, having taken want he wanted, to have simply dismissed the prisoner with no further thought, even though searching him properly and interrogating him might have actually revealed useful information.