Is it possible to run an exchange where the exchange operator would not have the ability to enter fabricated orders or to know the positions held in customer accounts?
Yes, sort of. I described a way to implement something like futures trading on top of colored coins. There are some limitations, of course. Particularly, contracts are capped, they aren't 100% same a futures. (Settlement exactly matches settlement of similar futures contract only if price is within a certain range. E.g. when BTCUSD is between 70 and 100.)
But otherwise, there is a lot of transparency. Basically, operator cannot do much without getting caught; he plays no role between launch and settlement; it is possible to get multiple parties involved, they would need to collude to go rogue.