I am quite certain that the gun pictured here is exactly what it says it is, a vintage 1938 prototype. There were no "later models" that did away with the barrel support sleeve.
But it has been "sporterized" by someone unknown, and seems to have been altered to remove that "sleeve". Whether that was done safely, I have no way of knowing, but I would certainly like to see the gun.
The Johnson is short-recoil operated. The barrel and bolt travel back locked together for about a half inch, then the bolt head is cammed out of engagement with the barrel and continues back to extract and eject the fired case. If the magazine is empty, it is locked back; otherwise, it comes forward to chamber the next round and the bolt head locks into the barrel.
So the barrel has to move and has to have support, which that "sleeve" provides. Also, of course, the close fit of the barrel in the sleeve is needed for accuracy.
As for a magazine, Johnson first used BAR magazines, so I think I would try that approach first and see if one could be modified to fit without, needless to say, altering the rifle.