Based on your description I'd be surprised if the two issues are related...
The thermostat relay issue is certainly something inside the thermostat itself. I've had some pretty squirrely problems with newer thermostats. Due to the increased functionality, the circuits and component parts are made much smaller and, I believe, less reliable. I'd replace the thermostat to solve at least that issue.
The blower coming on could also be a problem with the thermostat sending the fan signal when it isn't necessary (bad relay or miswiring) It also might be a fan timer or fan control issue with the furnace itself, although If it were the fan control I would expect the symptoms to appear a little less predictably.
It's also possible that it is normal... There are a couple reasons that a furnace blower might come on by itself, especially right after a call for heat ends.
* The fan could be activated by a temperature sensor and these sensors grow less accurate over time so it may shut off the fan prematurely and then as residual heat radiates from the heat exchanger it reactivates the fan for a second (third.. etc) cycle.
*an airflow restriction like dirty filters or dirty ac coil could cause the heat exchanger to overheat and after the blower shuts off it could be activating a temperature limit control which typically would turn on the blower and exhaust motors.