Update on situation!
@INTP, not sure what do you mean...
I added other 10k pullups, didn't help... The board has its owh 4k7 pullups so I think that should be fine..
Then I disassembled the display to be able to troubleshoot more, and I was able to detect i2c display through serial - confirmed 0x3F. Everythings run smoothly, the LCD backlight is solid on and I see the "no connection" blank line. Display is drawing 3mA.
But, here are things getting weird... I upload the basic test check sketch from https://arduino-info.wikispaces.com/LCD-Blue-I2C and change the address to 0x3F and remove the backlight flash loop (You'll se why). After arduino reset I am not able to see anything in the serial and after a second or two the backlight turns off and the back power LED seems to be very very dimmed. Display current drops down to 1.4mA. I can still see the line in the display, but the serial window is empty, and if I write sth into it, Arduino RX led flashes but nothing happens.. Only when I press the reset button, the cycle repeats (Display lights up for 2secs, then freeze...)
I hope this will be little bit more helpful for finding the answer, thanks for helping me so far 