Unit Impulse : Bounded or Unbounded?
Bounded or unbounded in that context always refers to amplitude. E.g., BIBO (bounded-input-bounded-output) stability means that a bounded input signal $x(t)$ satisfying $|x(t)|\le A$ always results in a bounded output $y(t)$ satisfying $|y(t)|\le B$ with some finite positive constants $A$ and $B$.

So the Dirac delta impulse is not well-suited for testing stability of a system. E.g., an ideal integrator's response to an impulse at the input is a unit step, which is of course bounded. Yet, the ideal integrator is unstable because its response to a step is unbounded.

In sum, a Dirac delta impulse is unbounded.