Dynamic analysis of cylinder on top of moving plate Consider the following setup

Everything is initially at rest. $P$ is a constant force which is then applied on the plate for a given time interval. There exists friction between the ground and the plate, and between the plate and the cylinder.
As you have probably guessed, this was part of an exercise, but what I want is to get some intuition on the evolution of the system. Below is my reasoning.
Everything is initially at rest. $P$ is an external force, so it will change the linear momentum of the plate. I know the friction force grows linearly until the maximum static friction force, and then immediately falls to the dynamic friction force. Therefore what determines if the system will move is the maximum static friction force between the ground and the plate, which is
$$ F_{fr} = \mu_s \times (m_{plate} + m_{cylinder}) \times g $$
If $P$ is greater than this, we will have movement. Otherwise, the system will remain at rest.
First question: Did I need to consider friction between the ball and the plate for this analysis?
Now that the plate is moving, the cylinder will tend to remain at rest.
Second question: How can I know the motion of the cylinder (I am more concerned with slippage) only from the data that is given?
The friction force that the plate exerts on the cylinder will be directed to the right. Therefore the cylinder will rotate counter-clockwise. Balancing linear impulse and impulse momentum I can write the linear and angular velocity of the cylinder in terms of the friction force. Then, applying the same technique to the plate, I can write the velocity of the plate in terms of the friction force between the cylinder and the plate.
Third question: Why is the friction force between the cylinder and the plate different from $\mu_{k} \times m_{cylinder} \times g$?
Fourth question: To reach the solutions, I equated the assumed the speed of the plate was equal to the speed of the contact point on the cylinder. Why am I allowed to assume this?

The actual values are the following, in case you find them interesting
$m_{plate} = 1.5kg$
$m_{cylinder} = 6kg$
$r_{cylinder} = 0.08m$
$\mu_s = 0.25$
$\mu_k = 0.20$
$P = 25N$
$\Delta t = 0.75s$
 A: *

*No you don't need to consider friction between plate and cylinder. Whether the plate moves depends on whether the applied force exceeds the limit of static friction between plate and ground. This limit depends only on the vertical forces acting at this surface. These forces are the same whether or not the cylinder moves relative to the plate.

*Adopt a frame of reference which is accelerating with the plate. This requires you to apply a pseudo force $-ma$ acting to the left on the CM of the cylinder. Friction will (partially) oppose this force and create a torque. If the limiting static friction coefficient $\mu_s$ is greater than a critical value, the cylinder will roll; if it is less than the critical value the cylinder will slide and roll. See Accelerated pure rolling and friction for details of how to calculate the critical value of $\mu$.

*If the cylinder slides, the friction force will be $\mu_k mg$. If the cylinder rolls and accelerates, static friction applies - so friction could be less than the limiting value $\mu_s mg$. Like normal reaction, static friction is a reaction force. It reacts to the pseudo-force by providing sufficient force to create a torque which generates rolling without slipping at the contact point, up to the limit of static friction. 

*I think your assumption is that the contact point remains stationary in the ground frame of reference. This is not a valid assumption. If the cylinder were a block sliding on a frictionless plate, it would accelerate to the left at $-a$ relative to the plate, remaining stationary in the ground frame. However, because rolling objects have greater inertia (compare objects sliding vs rolling down an inclined plane) the acceleration of the CM of the cylinder rolling to the left is less than $a$.  
See also Understanding rotational motion on a rough surface which addresses acceleration on an inclined plane, but the issues are the same.
