HC-05 bridge using Arduino Nano and GNU Screen
Based on the above comments I provide here the full answer in the hope it will be useful in the future.

HC-05 has two modes. The Communication Mode and AT Command Mode. In the latter mode, HC-05 responds in kind of "config" way and many default settings of the communication mode can be change here. While the baud rate of the communication mode, and many other settings can be changed, the baud rate of the AT command mode is always fixed 38400 to avoid any confusions and mistakes. On the other hand, the default factory baud rate of the communication mode, i.e. before any change, is set to 9600. So the solution to my problem was to set the baud-rate to 9600 on my newly purchased HC-05 and I was able to make an echo bridge between HC-05 Bluetooth and serial. Here is the code:

/*
 * echo script
 */

byte incomingByte;

#include &lt;SoftwareSerial.h&gt;

SoftwareSerial BTSerial(3, 2); // RX | TX

void setup() {
    BTSerial.begin(9600);
    Serial.begin(9600);
}

void loop() {
  if (Serial.available()) {
    incomingByte = Serial.read();
    BTSerial.print((char)incomingByte);
  }

  if (BTSerial.available()) {
    incomingByte = BTSerial.read();
    Serial.print((char)incomingByte);
  }
}


After pairing HC-05 with OSX, I was able to open two OSX Terminals with one of these commands in the first terminal:

screen /dev/cu.wchusbserial1410 9600


and in the other:

screen /dev/cu.HC-05-SPPDev 9600


Everything worked properly, i.e. typing in one terminal produced characters in the other and vice versa. For changing the default baud-rate in the communication mode, I found a very good tutorial which can be found here.