It's a lot of data, and virtually all of it, virtually always, indicates normal operation. There is no Internet service up in the air, except via satellite. That is pretty expensive, compared to ground installations, because the antenna is mounted on a moving vehicle, and it has to track the satellite. It's not exactly trivial.
There are radio reporting systems (on Airbus) that use HF, as I understand; that's how the crash of a Brazilian flight was investigated. The amount of data that you can reliably send over a weak HF link in arbitrary propagation conditions is measured in tens of bytes per minute. That's why only abbreviated alarm codes are sent this way, and only when there is an alarm.
Speaking of satellites, aren’t there cameras up there watching everything...all the time?