That ~2000 RPM speed you see is the pump's speed, which should be ~2000 RPM. Changing the H80's fan speed with the button has no affect on the pump speed, it is constant all the time. You can't monitor the H80 fan speed, since they are connected to the H80's fan controller and not to the mother board. From your post it sounds like that is how your H80's fans are connected. Some users connect the fans to a separate fan speed controller, or even the mother board, but that introduces other issues that must be dealt with.
On any PC fan, one side has three or four mounting braces or arms, that connect to the fan's motor. That is the side where air moves out of a fan, and is a 100% standard for fans. Some fans will also have arrows on the narrow sides that show the air flow direction, but others don't.
Any H-series cooler can be configured so the fans pull air into the case from outside, called intake, or push air out of the case, called exhaust. The fans are just turned one way or the other to do that. Both ways work and one style or the other is not wrong. Corsair usually recommends the fans be configured as intake, pulling outside air into the case. But, if a case does not have any other exhaust fans, such as mounted on the top of the case to pull the warm air from the radiator out of the case, it's better to configure the fans on a H-series cooler as exhaust.
It sounds like your H80 has been set up in exhaust mode. Air comes into your case through any vents in the case, and from fans usually on the front of the case. It's hard to feel air being pulled into a case through a vent, or even by a fan.
I doubt that your H80's fans are configured wrong, if you can feel air being pushed out of the case. To check the fans orientation, first look through the vent on the rear of the case, you should see the frame I mentioned. Next, open the case and look at the fan on the rear of the radiator, you should only see the fan's blades, and no frame.
The noise issue is not easy to deal with. If the fans change speed when you press the button, then the controller is working correctly. Is exhaust mode noisier than intake mode? Hard to say, and when the fans are in intake mode, you need more fans to pull air out of the case.