Why is the input trigger pulse in a monostable multivibrator fed to the base of the transistors through a capacitor and not an inductor or resistor?
&quot;Inventing&quot; the CR circuit
Necessity. Circuits with avalanche-like positive feedback like this (timers, triggers, latches, etc.) must be controlled with short pulses which only start the process but then leave the circuit alone. Another example from our daily lives is the staircase timer which we turn on with a short press. In all these cases, we have to shorten the continuous input signal (pulse).
C differentiator. For this purpose, we need an element that initially passes current but interrupts it after a short time. Such an element is the capacitor. While it is charging, current flows in the circuit;  but after charging to a voltage (almost) equal to the input voltage but with opposite polarity, it interrupts the current in the circuit. The same trick is used in bootstrapped circuits where the current is stopped by inserting an opposing voltage source (in another answer, I have summarized this trick into a &quot;thirth golden rule for stopping current&quot;:) See also my RG question about the capacitor behavior.
CR differentiator. Now all we have to do is connect a resistor in series to the capacitor to convert the current into voltage and our &quot;shortening circuit&quot; is ready. So we actually &quot;invented&quot; the differentiating CR circuit in an intuitive way, instead of taking it for granted as it appears in textbooks.
Implementation
A typical implementation of this idea for the case of the OP's circuit is shown in the picture below taken from Electronics Tutorials:

Operation
One strange feature of the differentiating circuit is that it produces a sequence of two short pulses with alternating polarity. This is because, in this configuration, the capacitor acts as a floating voltage source (a kind of a &quot;rechargeable battery&quot;).
While the input voltage is high, the capacitor is charged with a positive voltage and the transistor is turned on. When the input voltage becomes zero, the positive capacitor terminal is connected to ground. The capacitor begins to discharge in the opposite direction and a negative pulse appears on the resistor. Since this is undesirable for the NPN transistor, a diode is connected to stop it.
Note that this circuit will work properly if &quot;zero input voltage&quot; means &quot;grounded input&quot;... not &quot;open circuit&quot;. Otherwise, the capacitor will be always charged.
Coupling capacitor
The capacitor here is the same coupling capacitor as in AC amplifiers you mentioned but here it is a pulse controlled while in AC amplifiers the input voltage is sinusoidal.
The difference between the two applications is that here the capacitor is fully charged and discharged while, in AC amplifiers, it stays constantly charged and acts as a &quot;shifting battery&quot;. This is because here the capacitor must pass the current (input signal) only for a short time, while there it must pass it continuously. As a university lecturer would tell to their students, &quot;Here the differentiating circuit has a small time constant while there it is large.&quot;
See more about the idea behind the coupling capacitor in my RG question.
Why not an inductor?
The inductor has an opposite behavior - while the capacitor in series passes the current in the beginning and stops it after a while, the inductor stops the current in the beginning and passes it after a while. It is an inconvenient element, but if you still have a special desire to use it, you have to connect it in place of the resistor (swap the elements). It does this by the same means - by opposing an equivalent voltage to the input voltage at the first moment.
So, in the first moment it will not prevent the input voltage to pass current through the base-emitter junction of the transistor but after a while it will divert the current through itself (first &quot;golden rule for stopping current&quot; from the link above:)
See more about the difference between the capacitor and inductor behavior in my RG question.
And why not a resistor?
Well, this can no longer happen because the behavior of the resistor does not depend on time. The single resistor in series will allow you to control the transistor without problems but you will have to do it with short pulses. Its resistance will have to be significantly less than R3 because a voltage divider is formed.