Dragging problem

This article discusses a scenario/arrangement whose statics/dynamics/kinematics can be understood using the ideas of classical mechanics.
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The scenario

Suppose a block $m$ rests on a fixed horizontal floor. The limiting coefficient of static friction between $m$ and the floor is $\mu _{s}$ and the coefficient of kinetic friction is $\mu _{k}$. A force $F$ is applied on the block in a diagonal direction at an angle $\theta$ with the horizontal. The questions are as follows:

• For a given angle $\theta$, what is the minimum magnitude of force needed to get the block to start sliding, and how is the acceleration of the block given as a function of $F$?
• For what value of $\theta$ is the minimum magnitude of force needed to get the block to start sliding as low as possible, and what is this minimum magnitude of force?
• For what value of $\theta$ is the minimum magnitude of force needed to get the block to keep sliding as low as possible, and what is this minimum magnitude of force?

Basic components of force diagram

Force (letter)	Nature of force	Condition for existence	Magnitude	Direction
$F$	external force being applied	given to us that it's being applied	$F$	angle $\theta$ with horizontal. The horizontal component is $F\cos \theta$ and the vertical component is $F\sin \theta$.
$mg$	gravitational force	unconditional	$mg$	vertical, downward
$N$	normal force	assuming that $F\sin \theta \leq mg$, otherwise the block will lift off the floor.	vertical, upward
$f_{s}$	static friction	no sliding	horizontal, opposite to horizontal component of $F$
$f_{k}$	kinetic friction	sliding	horizontal, opposite to direction of slipping. Assuming initially at rest, direction of slipping = horizontal component of $F$, so this is opposite to horizontal component of $F$.