Pseudo Force Direction 
A cork and a metal Bob are connected by a string as shown in the figure.
  In case the beaker is given an acceleration towards left, the cork moves towards ..?

I answered that the cork moves towards right because of a pseudo force but the answer key says that it moves towards left due to a pseudo force. But if you look at it, it seems intuitive that the pseudo force  gets applied to the right, isn't it?

 A: Consider the more familiar arrangement of the mass $m$ on the end of a piece of sting with tension $\vec T$ (simple pendulum arrangement) which is accelerating to the left $\vec a$.
The system can be analysed by either seeing what happens in the inertial laboratory frame or in the non-inertial frame which is accelerating at the same rate as the string and the mass.
In both cases the bob moves in a direction opposite to that of the acceleration $\vec a$.  

In the inertial frame there is a net force $\vec F$ on the mass to the left which causes it to accelerate to the left $(\vec F=m\vec a)$.
In the non-inertial frame it is a static equilibrium situation with zero net force on the bob as a result of the pseudo force $ma$ acting on the bob to the right.
This analysis is fine in a vacuum and in air if the assumption is made that the density of the bob is much greater than that of air.
If that is not the case then two "upthrust" forces $\vec u$ and $\vec v$ must be included as shown below.

So this is what happens when the density of the bob is greater than the fluid it is immersed in.
The "upthrust" will be less than the "weight" of the bob.
The "upthrust" to the left is a real force and results from a pressure difference between the air to the left of the bob and the air to the right the bob.  
Now consider the example given by the OP.
Here the density of the bob (cork) is less than that of fluid surrounding it (water) and so the "upthrust" is greater than the "weight" of the bob.

You can see that the tension in the string is in the opposite direction to that of a metal bob in air and the bob is ahead of the point of suspension.
The upthrust on an air bulb can easily be shown by placing a builder's sprit level on a flat surface and moving the spirit level to the left.
The air bulb moves to the left on the centre line of the spirit level.
A: The cork floats so it will be to the left.
The force acts on the whole system, including the water. I think the easiest way to think of it is as a still reference frame and the pseudo force adding to gravity. Then imagine the new container with gravity vector pointing down and to the right and therefore the cork will float up and to the left 
A: Pseudo-forces in this case are due to inertial effects that the cork experiences.
Initially, the entire system including the cork, beaker, liquid, metal and string are stationary.  According to Newtons First Law of motion, these objects at rest will want to stay at rest unless a force is applied to them.
When you accelerate the beaker to the left, everything in the beaker initially has no force acting on it, and wants to stay at rest, but the beaker wants to move, and everything contained in it has no choice.  This means the beaker will apply a force on the fluid and on the metal ball.  These will resist, but the resistance will not be as noticeable, as the metal ball would require a decent force to move relative to the beaker, and the movement in the fluid itself wont be very visually apparent. 
The cork still wants to stay in place too.  As everything starts accelerating to the right, it wants to stay put.  From the perspective of the beaker, that is the same as the cork going to the left.  As the metal ball moves and the cork doesn't, this increases the tension in the string.  Eventually this tension is enough to get the cork moving with the same speed as the system, but at that point the string will be at some angle which makes the cork to the left of the metal ball.
