but apparently I have a problem in communication between the MC and the Bluetooth. I am not sure how to test if they are communicating or not.
I have only yesterday succeeded in getting some joy with Bluetooth.
I am not familiar with yours but I assume it is similar and they all work much the same way.
It seems that there isn't much to do between the Arduino and the bluetooth module. It is either connected properly or it isn't, and you will be amazed how often the latter is the problem. So check yours, twice.
There is no way of using the debug monitor to check that data is being sent via bluetooth. I understand Arduino doesn't distinguish between bluetooth and the serial monitor.
Seeing a flashing LED does not guarantee that it is ready for communication. It can do that with the Rx/Tx wires the wrong way round.
Similarly, having the PC discover the bluetooth is no guarantee of communication. It simply means the power is connected, something you probably already know. I guess it also means it will communicate - eventually.
I was close to giving up but it turned out that all the work was at the PC end, and I just needed a bit of close dedication. I use RealTerm set at 9600 8N1. You can use a desktop icon that runs a comprehensive command line so that it will work the same way every time.
For test purposes, I simply used a clock that normally printed to the monitor - no changes needed.
I don't know anything about two-way comms, but RealTerm can do it. I imagine that once you can receive, sending will be easy. Bluetooth has now become a mainstay for my projects, It is simple cheap and good for the occasional short dense bursts of data logging.