Should HVAC registers be closed or open when HVAC is broken, & window AC units are in use

Yes, placing registers (or other heat sources such as radiators) near exterior windows and doors is the usual practice. This is done in order to combat cold drafts and ensure a more even temperature throughout the room. Here's a Q&A on the subject from Ask This Old House:

Window glass is the coldest part of a wall. When warm room air hits it, the air cools, and cool air sinks. The movement of cool air creates floor drafts that most people find uncomfortable. The placement of forced-air heat registers or baseboard heating units under the windows counteracts this process by sending up warm air to mix with the cool. The end result is that the room feels more comfortable.

All are true! Generic questions receive generic answers. The advice about closing the upper or lower with heat or cool respectively follows physics of "hot air rise" so sucking up the hot air you just produced isn't very efficient. On the other hand, you don't want to block an intake if it would put additional load on your fan and isn't allowing the amount of air the handler is expecting as they are engineered to a spec to try and optimized HVAC and/or energy performance. But you have two returns and they are designed to be able to shut, so why would they do that. The vent may have been installed because they were cheaper/on-hand, because they looked better, or actually because they were intended to be shut. They also could have been installed correctly and to spec of the air handler but then the air handler was replaced. With so many variables (generic question) you will hear a lot of different answers and most people with lean on the side of caution.

So how to determine what is right for your situation, I've had a few scenarios where people want to place a table or open entertainment center near or in front of their return so I'm going to give you the same advice. Limiting the handler is bad; however, short term is negligible so we'll perform a before and after test. First I inspect the handler to make sure the fan blades are clean. Try and find the vent furthest away from the handler and tape a piece of light cardboard/posterboard or paper to the top; create a hinge, tape on top with paper covering vent. Without any obstruction, both returns open, turn the fan on and see how far the board/paper moves; repeat with a return closed. On a house I typically only check one vent but you can check more. Typically when there is a problem it is extremely noticeable so I'm not breaking out rulers to measure the distance.

If you have the one return closed and your handler is running longer or more frequent that is also a sign that you either need service, possibly just a new filter, or that you don't have enough air flow anymore.