What is the name of the process by which rising air cools, then sinks and spreads out to form cloud cover?

I think you might be talking about the process of convection. Here’s a useful way to thinking about it: on a very basic level, clouds are essentially formed in two major ways – vertically and horizontally. When a cloud is formed “horizontally,” two air currents with different temperatures collide head on, like two cars crashing. When two currents collide, the warmer air is pushed up and over the cooler air, because warmer air is less dense. If the cold air hits warm air, the warm air is pushed up and can form thunderstorms. If warm air hits a cold air front, however, it slides more gently over the top of the cold air and is slowly cooled, creating stratus clouds. (This whole process can also happen when an air current hits a mountain. It’s forced up and over the mountain, cooling as it rises and forming cloud cover and rain on the other side of the mountain.)

That’s the “horizontal” approach. The “vertical approach” is through convection. Convection is heat transfer through the motion of a heated fluid – like air or water. The molecules in cold air are closer to together, making it more dense than warm air. Gravity pulls cool air down past the warm air. As a result, warm air is lifted up into the atmosphere where the temperature is lower, causing the warm air to cool down. The process repeats. When air cools, some of the water contained within it condenses into drops that appear as clouds (and/or later as rain). Cumulus and cumulonimbus are common cloud types formed by the convection process.