Hello everybody
I found a really cheap 2x16 LCD module with HD44780-like controller.
This is the datasheet which clearly explains it can be used with 4/8bit mode.
http://www.datasheet4u.com/download.php?id=626872
I can't understand if you can use both two lines while using 4 bit mode.
Which arduino pins would be free if I use 8 bit mode?
You can use all the features of the LCD in both 4bit and 8bit-mode. I can't see any reason for using it in 8bit-mode, it will waste four pins on your Arduino.
Thank you Skumlerud for the quick reply 
I'm a total n00b in these things... but according to the datasheet (PDF page 13, Function set) it seems the one/two line display bit is sent to the DB3 line, which is not connected when using 4-bit mode.
But in the example it seems it sends the first 4 bits and then the last 4 bits to set line/character mode.
Do I have to manually write the code for this or it's taken care by the libraries for LCD modules?
I'm a total n00b in these things... but according to the datasheet (PDF page 13, Function set) it seems the one/two line display bit is sent to the DB3 line, which is not connected when using 4-bit mode.
But in the example it seems it sends the first 4 bits and then the last 4 bits to set line/character mode.
That is correct. When in 4 bit mode, you send two nibbles for each byte.
Do I have to manually write the code for this or it's taken care by the libraries for LCD modules?
If you use a library (like Lcd4bit) all of this is taken care of.