35XXX, prewar, so it's a so called "Early Small Ring" with long extractor. Assume it's German, if not, everything is possible. Initial feeling was rocker plunger was not functional so the trigger did not disconnect after firing. But you mentioned the weird behavior of "half-cocking" which confuses big time. What was blocking the hammer from dropping?? On this thing, only two pieces of steel could block the hammer, safety level and sear. Temporily put sear aside, because sear does not offer "half cocking" function.
In normal case, if safety level is pushed up, the hammer cannot be opened at all. Or, hammer already cocked, then safety level at up position will block hammer from dropping. But this one is hard to say because it's not an original gun. Anything is possible. Was the safety on?
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[Edit] If you post a picture of the gun, probably we can help with some clue. Otherwise, it sounds like a 1930.
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