Do the actions the Prophet (PBUH) refrained from fall under the category of bid'ah (innovation), makrooh (disliked), or muharram (forbidden)? Or does that depend on the action itself and its desirability? And what is the criterion for that?

The Prophet (peace be upon him) leaving an action in the realm of acts of worship becomes a commendable Sunnah if there was a reason for the action and no impediment prevented it, otherwise, the action becomes a blameworthy innovation (bid'ah). As for the realm of customs, the default is permissibility, and there is no harm in doing something that the Prophet (peace be upon him) left.