dnd 5e – Does Simulacrum copy the skills and class characteristics of the original?

… has half the maximum hit points of the creature and is formed without any
team. Otherwise, The illusion uses all the statistics of the
creature that duplicates

The statistics of a creature include all its abilities, characteristics and abilities according to the extensive list in the monsters manual. In that list, the skills are under "Skills" (page 8) and the characteristics are in "Special features" (page 10).

Jeremy Crawford also confirms that the features and abilities of the class are considered "statistics" for a PC here:

Q: A magician retains all of his launching abilities, class characteristics, endowments; after launching True Polymorph (Planetar) on itself?

A: The true polymorph replaces the objective game statistics with
Those of the new creature. Objective characteristics are not transferred.

Also, if a drill did not inherit the characteristics of a creature, there would be no way for it to gain the ability to cast spells, for example. But we know that it can because the spell says:

nor can he recover the spent spell slots

So, yes, the simulation you created will have class characteristics and abilities of the creature that it duplicated.