To start with, I got into a pickle with newLiquidCrystal I2C libraryand moved the original related libraries out.
Next I tried the MCP chip using Adafruit's I2C library, and that worked fine.
Then I found your comprehensive hd44780 library and started working through the examples - a bit of a learning curve as my experience of these backpacks is limited, but most of the examples worked out of the box.
What hasn't helped so far, is that though there are many tutorials on the subject, a lot are light on the pinning information front and apart from Adafruit and Sparkfun, a lot of the commercial vendors a understandably coy with schematics. My take on this is that I'm not going to buy another commercial unit until I've given the chips a shot.
If you tell me what you want to look at, I'll do my best to provide. The hardware is Nano board (328P but with CH340 instead of FTDI - i.e. a decent clone), PCF8574A DIP-16 and Midas 16 x 2 panel with full set of 16 pins (soldered by me), contrast pot (10k, working), 4k7 pullups on SDA and SCL to A4 and A5 respectively. All wiring is wire-wrap and checked end-to-end, but were're not infallible!
by the way, the only one or two examples of yours that didn't work as expected, were those associated with newLiquidCrystal examples.
Somewhere along the line, I recently added a new library called Arduino-LiquidCrystal-I2C-library (how confusing is this) and tried the "Hello World" example. It didn't work. However, there is a configuration line (address, pins etc.) in an example in the link above from mjkdz Courtesy of PeterSek which I substituted and that worked straightaway.
const uint8_t addr = 0; // zero means auto locateconst uint8_t rs=0, rw=1, en=2, d4=4, d5=5, d6=6, d7=7;hd44780_I2Cexp lcd(addr, I2Cexp_PCF8574, rs, rw, en, d4, d5, d6, d7);
The only thing that didn't get to work was the little code snippet with the manual configuration.I popped it into the USER CONFIGURABLE section and commented out hd44780_I2Cexp lcd; // auto locate & configure up to 16 displays as it was coming up with declaration conflict.