So I do not need any special equipment on PC side right, I can even connect to the ESP's AP with mobile, right?
If you set the ESP8266 in AP you can connect to it with every wifi capable device.
You can control an ESP8266 with just an Arduino, but you I would not recommend using your Arduino as the power supply. If I remember correctly the mega's VCC pins are rated for 200mA which may or may not be enough for your ESP8266 I'm not sure, but I would recommend using another power supply for the ESP8266.
Is it possible for the ESP to be WIFI client and connect to regular WIFI SSID
Yes that is possible.
Is it possible to communicate bidirectional somehow?
I am not exactly sure what you want, but it is possible to communicate in both directions. Assuming you know the local IP address of your ESP8266, you could for example send an HTTP request to it using any browser to which the ESP8266 can reply with anything.
Your ESP8266 can also send data to your computer, but it would require some program on your computer listening for incoming connections. Also possible, but more difficult.
At last I would like to tell you that driving an ESP8266 with an Arduino is like driving a Ferrari with the engine from a tuk tuk. It will move, but you will never use all of it's features. I don't know which version of the ESP8266 you have, but there are versions out there that can be programmed via USB just as your Arduino and you can just use the Arduino IDE and all of it's features like the Arduino language.