NSEU,
Would the ALTN circuit be latched and ineffective (until cycled off/on)?
Yes.
In this condition, with the gear up, switch still selected and then the gear lever put to OFF (normal flight position) would the gear and doors stay up?
Yes; the system is back to it's mechanical state.
I was thinking the lube cover around the aileron programmer but I don't think it's that big; it's nearest thing to the gear that I can think of that could get dragged down by a burst tyre (if that was the cause of system 4 loss).