Is it possible to in-flight receive navigation data on an electronic device (like smartphone and/or tablet) present in the cockpit? I refer to General Aviation aircraft and navigation data like airspeed (not groundspeed), GPS and AHRS data.
Garmin has a product called Connext that allows you to stream in-flight avionics data to mobile devices. They are pushing that product as a move towards connected components and a "Bring your own device" approach to cockpit automation.
So is it possible in general to receive it? No, not really. Not unless the avionics package was designed to broadcast it out. As other answers have said, a simple GPS and AHRS integration of the sensors in your tablet/smartphone can give a pretty good approximation of the data you are asking for, but things like airspeed, navigation waypoints, and other on-board sensor data are not generally available without a product like Connext.
The other side of it is that if you are a passenger you still probably won't be able to pair with the avionics. Connext allows you to modify the flight plan via a mobile device, and I'm not sure I would want any of my passengers doing that. On top of that, pilots are trained to interpret the data at hand, you don't want a passenger seeing a faulty artificial horizon after a vacuum failure and think that the aircraft is tumbling out of the sky.
There is a lot of security concerns as it is over being able to hack into commercial aircraft (I know you are talking about GA), but I still don't want a passenger to have the ability to connect to my avionics.
EDIT: This answer applies to large commercial aircraft.
As a normal passenger? No. As @mins said, this data is captured, processed, and displayed within the aircraft's avionics system, but this data is not available to passengers (it's not transmitted outside the avionics system).
When aircraft are being developed and the manufacturers perform flight tests, then they will install special gateways into the avionics, allowing the test engineers in flight and on the ground to monitor aircraft parameters outside the cockpit - but this equipment is removed long before aircraft are sold to an airline.
The best you can do is to use your own sensors - e.g. the GPS on your phone, a separate accelerator/gyro module - to generate your own position and attitude data. However, it will have a worse accuracy than the aircraft-level sensors, and there are certain parameters (e.g. airspeed) which you won't be able to obtain.
Yes but only with certain avionics. Avidyne recently released an SDK that allows third party developers (tablet/smart phone) to interface with their IDF440 and 540 units. There are two levels of access (read only and read/write) that an app can have for the data stream.