I'm very sure that this is the Lunar Equipment Conveyor.
The Lunar Equipment Conveyor (LEC) is a device which the astronauts will use during the EVA to transfer equipment to or from the ascent stage. It may also be used by the crewmen as a safety tether when moving down the ladder or as an aid in ascending to the ascent stage.
The LEC is a 60 foot continuous loop of 1 inch wide strap, which loops through a support point in the ascent stage and back to the crewman on the surface. The end of the loop is closed by two hooks, attached together
- What material was it made from? Were there problems with outgassing or UV damage that had to be addressed?
All of the references call it "webbing" or "strap material". OrganicMarble says that similar straps on the Shuttle were made of Nomex, which is most likely what was also used in Apollo. Aircraft seat belts (MIL-SPEC W-4088) are made of nylon. Here are two ads for such webbing; the latter specifies a strength of 5000 pounds at 2-inch width. Automobile-grade seat belts are made of polyester.
- Was it just a length of cord with free ends or were there hooks or eyelets on the ends?
There were carabiners (hooks) on each end. They were intended to be connected together to form a loop 30 feet long.
The loop went over a pulley, which hung from the "PLSS Upper Donning Station". The latter was a set of yellow bars which hung from the ceiling inside the LM cabin.
- Was there a preferred type of knot that worked well for this type of cord and could be easily tied by a fully-suited and pressurized astronaut's gloves?
No knots were intended; the carabiner hooks were enough to attach to whatever they needed.
- Was it ever used on any Apollo mission, either to haul someone out of a crater or for something else?
The Gemini program used a similar device called a "waist tether":
On each of the last three missions, the pilots who went outside had complained that they needed more help in body positioning. Each spacecraft carried more restraints than the one before. The 9 restraints on Gemini IX-A had become 44 on Gemini XII. One helpful innovation was a waist tether that allowed the pilot to retrieve packages, turn wrenches with considerable torque, and attach the vehicle tether without undue stress.
On the Shoulders of Titans, section 15.4
Waist tethers were even used during Gemini XII to hold two spacecraft together!
The next day Lovell and Aldrin got ready for the main event of the mission - see if a man could perform useful tasks in space at the end of an umbilical. Near the 43-hour point in the flight, Aldrin stood up in his seat and reinstalled the movie camera just as easily as before - then removed it, stepped into space, and replaced it, using only a handrail to maintain position. The astronaut then moved, hand over hand, along the rail to the nose of the Agena docking-adapter. Using his waist tether for restraint, he tied the two vehicles together for the gravity-gradient experiment without any of the problems Gordon had encountered.
Planning for Apollo recognized two needs for tethers:
first, as a conveyor device; second, as a contingency lifeline for EVA return to the CM, following a hard-dock failure.
Apollo Experience Report: Crew Provisions and Equipment Subsystem, NASA Tech Note D-6737, p. 22
The first need -- a conveyor device -- was satisfied by lengthening a waist tether and adding a pulley. The result was the LEC. It was tried on Apollo 11 and 12, but dust was found to be a problem. It was deleted in later missions.
Initially, a pulleylike double-strap conveyor was used to lower equipment to the surface and raise it into the cabin. The Apollo 11 crew found that, when the straps became heavily coated with dust, the dust fell on the suit of the surface crewmember and also deposited in the lunar module cabin. The dust ultimately seemed to bind the pulley so that considerable force was required to operate the conveyor. A single-strap conveyor was used for Apollo 12 operations , but the crew reported that this conveyor also collected dust which was subsequently deposited in the cabin. In lieu of using a conveyor system, the Apollo 14 crew reported that stability and mobility on the ladder, maintained by using only one hand for support, seemed adequate to allow carrying equipment up the ladder.
Apollo Program Summary Report, p. 6-8
Waist tethers continued to be used during Apollo for the second need -- an emergency EVA from the LM to the CM. This was tested on Apollo 9. Two were carried on each mission.
Because he had no idea yet if it was safe to step on the moon, Neil Armstrong also used the LEC and a waist tether as a safety line when descending the ladder. In the picture of training below, the brown line running down the ladder is the LEC; the waist tether is clipped between the LEC and Armstrong.
Armstrong used the other waist tether to hold up his feet while he slept in the LM. After Armstrong died, his widow found a bag of equipment in their closet. One of the items was this second waist tether.
Neither type of strap was used to rescue people or equipment.
So Apollo 12 did have a 30-foot tether, but it wasn't a rope.