No that's not how it works at all. Compressed air is bled from the engine compressors (or 2 electric compressors in the case of the 787), cooled down using ram

No that's not how it works at all. Compressed air is bled from the engine compressors (or 2 electric compressors in the case of the 787), cooled down using ram air, and passed into an air cycle machine that uses ram air to get the desired pressure and a slightly low temperature. Then it's mixed with filtered recirculated air (taken below the floor around the center of the cabin). Then it's distributed by zone into the whole cabin, brought up to the desired temperature by injecting hot air from the beginning of the cycle, and delivered by nozzles all along the cabin. And excess cabin air is bled out of the cabin, which maintains the right pressure. However in the case of the virus, the biggest risk arguably doesn't come from the recirculation but the nearby passengers.