They suspected there was a free water passage because the coast was well known on both side. But the interior of the land was terra incognita, and that was the point of the expedition : finding the passage and going through it if it existed.

However, when Roald Admunsden crossed for the first time the North West passage a few years after the Franklin expedition was lost, it was not by boat, but with sleds...

From my understanding, given the size of the Terror and Erebus, the Franklin expedition could not have crossed the NW passage anyway at that time.

Because the route was occasionally navigable, dependent on season and weather; but if trends continue, the route will be consistently navigable.