- Right Click to Open the Shulker;

In that case the priority will be on the placed shulker, so this will be opened instead of the shulker in the hand.
Yes, any inventory shurtcut that you use in a normal shulker can be used in this.
Yes, shulker's names are one of the best underappreciated features on shulkers.
Of course, only Shift Right Click places the block
No, this is singleplayer only.