Yes it’s possible. Not all aircraft transmit the same data though, and it depends what software you are using as to whether you can see it or not.
For example, in the map display of modesmixer2, if you select a flight you get a pop up window that gives all the available data. Looking at an Easyjet A319 flight, I can see the following data:
ICAO, Squawk, Track, Altitude, Speed, VSpeed, Lat/Lon, True Airspeed, Indicated airspeed, Mach number, Ground speed, Magnetic heading, Altimeter pressure setting, and FCU set altitude (this is the one referred to in the Germanwings flight). There is also weather data such as wind speed and direction and the temperature.
There are several other fields for parameters relating to the flight management system, but for most planes they appear blank. The exact parameters transmitted seems to vary between aircraft types and airlines.
I’m not sure what other software lets you see that data - Virtual Radar Server has quite a lot of fields you can add to the detail panel, but it doesn’t seem to include all those given in modesmixer2.