TL;DR
- its friction that is responsible for the COM's acceleration.
- No.
Earlier on $\ldots$
This is an interesting question. As formulated right now - a vertical water tank that has sprouted a leak in its wall- it is not evident that even though in the absence of gravity the COM wouldn't move, gravity itself isn't important to the spirit or solution of the question.
Indeed, in an earlier answer, I had wrongly argued that the star of the show - friction on the tank - plays no role. And I had good reason to. (see the detailed rebuttal at the end). I had ignored the knot in my stomach that was forming the moment I had accepted that a vertical force could lead to horizontal acceleration; its a fluid so maybe $\ldots$
New perspective
Let us simplify your system without loosing any of the physics. Let us replace the leaky water tank with a clamped or constrained rocket, at rest wrt. lab . The water is now replaced with the exhaust gasses.
We see that while switching off friction (or clamping, in the case of rocket) leads to the same behavior of COM in both cases - no movement - switching of gravity doesn't : while the water doesn't move, the rocket's exhaust is unaffected. This shows that gravity only plays the role of providing the kinetic energy to the water plume. Even thought it was a vertical external force that provided the energy, it need not necessarily be so, as in the case of the rocket, where exhaust energy could have come from say, burning fuel.
So the rocket accelerates some gas and the gas accelerates the rocket back equally and oppositely, right? Only in the frictionless case.
In the 'friction-full' case , the rocket (tank) is clamped. So the gas pushing of the rocket is the same as pushing off the ground - the constraint force immediately pushes back keeping the rocket stationary and the gas accelerating.$^1$
Since the COM moves with the accelerating gas, its friction which is responsible for the movement. Strangely enough, in this sense the acceleration of the gas or water is no different than ordinary walking, where one similarly pushes of the ground, on earth or on moon.
Another line of thought which leads to the same result and would have avoided the pitfalls of reasoning in terms of the internals of the system is the blackbox argument - when in doubt use a blackbox.
Lets put the entire rocket/tank inside a blackbox. Since the COM is moving, we see the box moving horizontally. The only horizontal force acting on the box is friction so it must be the cause for the COM's horizontal acceleration. (Why its acting at all - only those inside the box know).
The second question
2: Is such type of motion of center of mass possible in which the components of the system don't move but the center of mass does ?
No. The COM is used to characterize the state of motion of the system as a whole. Its the average of their motions. If every component of a system is at rest wrt. a lab frame, so would be their average and hence the COM.
Rebuttal of the earlier answer
In my previous answer I had given a similar case involving force redirect and COM movement resulting from gravity:
Consider the case of a mass hanging from a rope on a pulley. The other end of the rope, going over the pulley, attaches to another mass but this one initially rests on a horizontal table. Upon progressing time, though gravity acts vertically on the first mass and accelerates it downwards, the second mass accelerates horizontally. Why is that?
I has used the above analogous scenario along with the following observation
Its a similar reasoning for the acceleration of the COM of a sprouting water body. The weight of the water column in the tank is responsible for generating a pressure in the bulk of the liquid (const. at a given depth). This pressure by Pascal's law acts uniformly from all directions at any given point. This nature of pressure makes it a "3D pulley" - it can redirect the force of gravity in all directions.
to argue that it was the redirection of gravity by gauge pressure that was responsible for the COM's movement. After all, if it could do it once, it could do it again.
What about the friction on the tank? Here's what I had said earlier:
$\ldots$ the exhaust of the water plume as it gushes out on the left under the gauge pressure from the right, exerts a force on the tank. Static friction counters this force. It therefore isn’t responsible for the shifting COM.
These three deceptively simple arguments had together convinced me that gravity was the protagonist of this problem. Alas, upon further (and tiresome) scrutiny, they fall apart:
In the case of the pulley, the net horizontal acceleration of the COM isn't provided by gravity. It is provided by the normal reaction at the pulley. This is most clearly seen in the blackbox argument as there is no other horizontal force in this system to begin with.
Yes, Pascal's law allows for a redirection of weight but it is still an internal force albeit sourced because of an external force. The fact that the same pressurization of fluid can be obtained using internal mechanisms bears out this point. Moreover, the pressure is hidden in the blackbox argument and so its gravitational origin is of no concern.
The fact that friction counters the exhaust's force on the tank is correct. But by the same reasoning, it also supplies the reaction on the exhaust. Since there is no other force on the exhaust, it must accelerate and take the COM with it, thus establishing friction as the driving force.
Appendix
$^1$ In fact both friction and the force on exhaust can be calculated to be $\rho v_{exhaust}^2 A$ where $\rho$ is fluid density, $H$ is column height, $A$ is orifice cross-section.
A.1 The man seems irrelevant to the discussion about the accelerating COM. "The COM moves from some initial value" - that alone is enough to inquire about the external force. Using an equal mass oppositely placed, besides bringing the initial COM to center, plays no differentiating role.
A.2 Thanks to Dale for motivating further scrutiny.
