Anomen's a joke, regardless of whether he passes that test or not. If he fails the test, though, he becomes (even more) mentally unstable, and starts randomly attacking party members for no good reason, so I'd slit his throat before he slits yours.
There's no time limit, nope.
And you can put your stuff anywhere, it'll never be stolen. I dropped the Equalizer in the middle of the slums, did two quests, came back, and it was still there. : P
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