Did the FAA order the installation of "Cooper vanes?"
The answer to both of your questions is "yes". The FAA ordered, in effect, the installation of "Cooper Vanes" and did do so in 1972.  

The reason you could not find the answer may have been that the modifications were not called "Cooper Vanes" by the FAA and nor was the FAA targeting the 727 specifically. Instead they addressed ventral and tailcone exits on any aircraft. 

CFR 14, Part 25 (Airworthiness Standards: Transport Category Airplanes), Subpart D (Design and Construction, Emergency Provisions) contains, since 31st of December 1972, the following text (emphasis mine):


  Sec. 25.809
  
  Emergency exit arrangement.
  
  ...
  
  (j) When required by the operating rules for any large passenger-carrying turbojet powered airplane, each ventral exit and tailcone exit must be--
  
  (1) Designed and constructed so that it cannot be opened during flight; and
  
  (2) Marked with a placard readable from a distance of 30 inches and installed at a conspicuous location near the means of opening the exit, stating that the exit has been designed and constructed so that it cannot be opened during flight


This rule was announced in the Notice of Proposed Rulemaking. Notice No. 72-15, Issued on June 6th, 1972.

It clearly references the hijackings as the reason for this change. 


  ....
  
  In spite of concerted efforts made by the FAA and the air carriers,
  incidents continue to occur wherein the safety of the flight of
  aircraft engaged in passenger-carrying operations under Part 121 of
  the Federal Aviation Regulations has been jeopardized by persons
  intending to harm the crew or take command of the airplane. On a
  number of occasions in recent hijackings, the ventral exit of an
  airplane has been opened and a hijacker aboard has parachuted from the
  airplane through that exit. The agency recognizes that every possible
  step must be taken to deter persons from boarding aircraft for such a
  hijacking purpose. Accordingly, the FAA deems it appropriate to
  propose certain amendments to Parts 25 and 121.
  
  Specifically, it is proposed to amend Sec 25.809 to provide that, when
  required by the operating rules, for any large passenger carrying
  turbojet powered air-plane an approved means must be provided so that:
  
  (1) takeoff cannot be started if either the ventral exit or tail cone
  exit is not locked; and 
  
  (2) neither the ventral exit nor the tail cone
  exit can be opened in flight.
  
  A similar amendment is proposed to be made to Sec 121.310, to become
  effective with respect to persons conducting operations under Part 121
  six months following its adoption.
  
  However, it is to be noted that to achieve compliance with the
  proposed regulation both the ventral exit and tail cone exit would
  have to continue to meet all of the requirements applicable to their
  approval as emergency exits. Specifically, to achieve compliance, the
  conditions that would have to be met to obtain approval of
  modification to the locking mechanisms of these two exits are as
  follows: 
  
  (1) The mechanism must be locked while the airplane is
  aloft;
  
  (2) Takeoff of the airplane cannot be started if either ventral
  or tail cone is not locked; and
  
  (3) The exit must be available for use
  in the event of an emergency.


Since Part 25 is for airworthiness certification of new aircraft, an additional change was proposed to Part 121 so that operator of existing aircraft were given 6 months to apply the new rules to already certified aircraft. 

Somehow the change to Sec 121.310 cannot be found in the FAA Regulatory and Guidance Library (at least not dated 1972), even though the final rule of 30th November 1972  includes it.

A revision of 121.310 from 1997 includes the text.



Special thanks to the FAA for making the pre-internet era rulemaking archives easily accessible through the internet.