How to check thermostat is working?
With that thermostat it would be easier to tell if the air handler and heat pump are working rather than the thermostat.  The thermostat doesn’t send a signal it closes switches just like a few automatic light switches. So you just need to see if the switches are closing. With that thermostat it may be a little difficult especially if it is not battery powered.
To test the air handler connect a jumper wire to the R terminal and to the G terminal. That will turn on the fan. To test the emergency heat connect a second jumper from R to W2. This would turn on the emergency heat. Remove the jumper from W2 and place it on Y/W1 to turn on the compressor.  Placing a third jumper between R and B will energize the reversing valve and cycle between heat and coil. Whatever you do, do notjump anything to C or at best you will blow a fuse and at worst let the smoke out of several components.  If all that works but does not work after reconnecting the thermostat it is most likely a thermostat problem. 
If the thermostat is battery powered you could use an ohm meter to tell is each switch is closing or not. Each one of those pins on the bottom go to a terminal. Find which terminal they go to using an ohm meter. Then insert test leads into the corresponding holes in the thermostat and do the same as above except instead of different pieces turning on your ohm meter will go from OL to 0