Is the referee allowed to show a yellow or red card to the coach? Is the referee required to issue a caution to a coach before dismissing the coach?
Law 12 doesn't deal with coaches at all. There is nothing in Law 12 about coaches. It only deals with players, those who have been players, and those who are eligible to be players. If irresponsible behavior is committed by anyone else in the immediate area of the field, the circumstances must be included in the game report.
If Law 12 does not give referees the authority to "card," does the referee have any authority to caution or send-off a coach? If yes, then which law gives the referee this authority?
Law 5 -The Referee ~ gives the authority to the referee to deal with inappropriate behavior on the part of coaches and team personnel other than players: The list of Powers and Duties of a referee in Law 5 includes: "takes action against team officials who fail to conduct themselves in a responsible manner and may at his discretion, expel them from the field of play and its immediate surrounds."
Note that nothing in the law requires a coach to be warned. Unlike players who dissent, coaches are not given a "second chance." There is nothing that requires a referee to "caution" the first dissent and then to send off if the dissent continues. Coaches can be dismissed the first time their behavior is irresponsible or brings the game into disrepute.
Many coaches today seem to believe that they can dissent with ever increasing volume until they get a caution. Then they will moderate the dissent somewhat so that they don't get a "second caution" and get sent off. But until they get that first caution, coaches seem to believe that they are immune from being sent off. Clearly this is not the case and, as referees, we need to enforce the law as written. If we do what we are supposed to do, maybe coaches will learn, clean up their acts, and improve the game of soccer.
What about competitions that involve youth teams where the referee is asked to show cards to the coaches? Should the referee show cards as asked by the rules of competition [that is, by league rules]?
A referee is an independent contractor who enters into contracts to referee for leagues and tournaments. When a referee accepts an assignment where certain rules of competition apply, one may argue that the referee accepts those rules of competition. If the rules would cause you to do something you believe is wrong or would cause you to violate what you believe to be the proper actions on the part of the referee, then the referee may elect to decline the assignment.
Your SRC believes that you should show a card as a supplemental action to assist leagues in controlling the actions of team personnel. (Emphasis added) If a coach's actions are irresponsible or bring disrepute to the game, dismiss the coach and then, if the league wants you to, show the card to indicate to everyone that the coach was dismissed. If the coach's actions warrant a warning of a nature comparable to a "caution" you would give a player, warn the coach and then, if the rules of competition require it, show the card to indicate to everyone that you are reporting the warning as a caution on the game report. Then, be sure to indicate on the game report that you dismissed or "cautioned" the coach.
What appears to be a good mechanic is to NOT display the card to the offending team official, but rather to display to card away from the individual but within clear view of the teams. To display a card to a team official who is already upset is likely to increase the probability of a confrontation. To deal with the misconduct, dismiss the offending team official, and then display the card will decrease the likelihood of confrontation and the problems that arise when confrontation occurs.