so I could take the lcd and battery and disconnect it from the arduino to carry the message around. I know this can easily solved by 2 arduinos but I just wanted to know if it can be done just with external power source. Thank you
Most have their own driver IC, and the Arduino just sends the data that has to be show on the display to that driver IC. No new data = no change. So removing the Arduino does not affect such a display, any info shown will remain.
As a rule of thumb: grounds should ALWAYS be connected, as that provides the reference potential (by convention called the 0V level). There are very few exceptions to this rule, and those usually revolve around the use of optocouplers and wanting to keep circuits completely electrically separated.