Could an internally tethered rocket cause external movement? Imagine a scenario where a large box is floating freely in space. There is a vacuum both inside and outside of this box. Also imagine that a dormant rocket is inside this box, but it is loosely tethered to all sides of the box. When the rocket fires, and tethers are pulled taunt, will the box move also?
 A: Briefly, yes. The box would speed up momentarily, until the propellant from the rocket hits the back end of the box and slows it back down again. Once the propellant is exhausted, the box will again be motionless (neglecting the minuscule jiggling caused by thermal motion of the propellant gas inside).
The controlling principle here is that the total momentum of the system is conserved; since in the reference frame you described it starts out at zero, it remains forever zero.
A consequence of that is that the center of mass of your closed system cannot move. Thus, once your rocket runs out of propellant, even though the box might have shifted over to the right a bit, the propellant on average will have shifted over to the left a bit, so the overall center of mass remains in the same place.
By the way, you might find it beneficial in thinking about this to eliminate the tether, and just affix the nose of the rocket to the right side of the box. It's a pretty similar situation, and conceptually simpler.
