Re: Do moon landings occur if possible at the dark/light terminator?

Using the data available below, it is possible to determine that all six successful Apollo landings occurred close to the terminator. (In order to determine the location of the terminator, interpolate between phases using the date and time of the landing.) All the landings were on the daylight side of the terminator, within about 25 degrees of it. So yes, it does appear that moon landings generally occur near the terminator.

Data on place/date/time of Apollo landings available here: National Air and Space Museum Apollo Landing Sites

Data on moon phases available here (enter year): U.S. Naval Observatory Phases of the Moon

The second part of your question asks why lunar landings occur near the terminator. I have not been able to find a definitive answer to this question, so all I can offer is a few educated guesses.

There are three potential reasons for landing near the terminator (or at least three I can think of): lighting, power, and cooling.

The first reason I thought of for landing near the terminator was lighting, and I still have a sneaking suspicion that this may be the actual reason the Apollo landers landed where they did (although see below). It wouldn't make any sense to land on the dark part of the moon, because the astronauts wouldn't be able to see what they were doing. Therefore, you would want to land somewhere where you would be guaranteed of having light throughout your entire stay. Landing just as the sun is rising at your location (in other words, landing at the terminator during the moon's waxing phases) would allow for sunlit lunar surface stays of as long as two weeks. (Of course, none of the surface missions was longer than a few days, and most of them were considerably shorter, so this argument might not hold much water.) For short stays, landing near the terminator at lunar dawn would provide for the greatest variety of lighting angles during the mission, since lighting of a scene changes dramatically in the time shortly after sunrise and shortly before sunset. This may have been a benefit to the photography carried out during the lunar landing missions.

The second reason I could think of for landing near the terminator would be power. Again, landing at dawn gives you the longest possible time in the sunlight at a given location, and the sunlight could be used to generate electrical power via solar cells. Unfortunately, all the Apollo missions (as stated above) were of such short duration that it would have made more sense to land when the sun was more directly overhead in order to allow the maximum solar radiation flux to hit the solar panels. More damningly for this argument, none of the Apollo missions used solar panels to generate electricity - they all ran on batteries.

The only other argument I can think of for landing near the terminator is that it allows for somewhat more efficient cooling. (This is the reason you stated in your question.) I think this is a very good argument, and it may have been the deciding factor in determining where to land spacecraft on the moon. Since there is no atmosphere on the moon, the temperature of a spacecraft or an astronaut (or anything else) cannot be regulated by convection as is normally the case on Earth. Therefore, any time the spacecraft or the astronaut needs to get rid of heat (for example, when they are being heated too stronly by solar radiation), this heat must be radiated either out into space or into another body. Radiative heat transfer is never very efficient compared to convection or conduction, but it can be made to work if the body absorbing the radiation is large enough and cool enough. On the surface of the moon, the largest, coolest heat sink around is deep space. If the spacecraft's radiators can be placed so that they can always radiate heat away from the surface of the moon and away from the sun, they could be made smaller than if they had to radiate into a warmer or smaller body. The easiest way to ensure that the radiators are always pointing away from the sun is to ensure that the spacecraft provides its own shade, and this is much easier to do at lunar dawn, when the sun is at or near the horizon, than it would be to do when the sun is overhead.

The above are just my thoughts on the matter, and as I said, I have not been able to find a definitive answer. Hopefully, this will give you enough to go on. If not, some research into old NASA materials or books on the Apollo program itself will probably help.