Do Federal regulations compel airlines to avoid schedule disruptions?
The Department of Transportation has A Consumer Guide to Air Travel that discusses passenger rights regarding delayed and cancelled flights. Some key points:

Airlines don't guarantee their schedules, and you should realize this when planning your trip.
Overbooking is not illegal, and most airlines overbook their scheduled flights to a certain extent in order to compensate for &quot;no-shows.&quot;
Airlines set their own &quot;boarding priorities&quot; -- the order in which they will bump different categories of passengers in an oversale situation.
DOT requires each airline to give all passengers who are bumped involuntarily a written statement describing their rights and explaining how the carrier decides who gets on an oversold flight and who doesn't.

There airline is required to provide various sorts of compensation if they bump you involuntarily and can't get you there within 1 hour of the original flight. This depends on how late you will arrive and what kind of ticket you have.
There is no rule that says airline employees must fly instead of customers, but each airline has their own Contract of Carriage that describes their rules. You can read United's here as an example. Rule 25 describes the process for denied boarding:

If there are not enough volunteers, other Passengers may be denied boarding involuntarily in accordance with UA’s boarding priority:
The priority of all other confirmed passengers may be determined based on a passenger’s fare class, itinerary, status of frequent flyer program membership, and the time in which the passenger presents him/herself for check-in without advanced seat assignment.

Of course IANAL but the way I read it, they &quot;may&quot; use any of those methods but they are free to deny boarding as they wish. Nothing stops them from choosing to bump customers before employees. Delaying or canceling a later flight because they bumped the employees would be more expensive than paying a few customers to be bumped involuntarily. But bumping employees may be less expensive than a PR fiasco.
There is also a program called Essential Air Service, where the DOT subsidizes airline service to smaller markets. Airlines bid to provide this service under certain conditions. Although I wouldn't say they are &quot;compelled&quot; to provide that service, if they don't meet the conditions, which could include not completing enough of the scheduled flights, they may not receive the subsidy.