Can one see from other program if Gedit is running and contains unsaved text?

Gedit is running

pidof gedit should give you the answer

contains unsaved text

As far as I know Gedit doesn't support any DBus interface which makes this task relatively difficult but possible:

#! /bin/bash

for i in `pidof gedit`; do
    for j in `xdotool search --name --onlyvisible --pid $i`; do
        title=`xdotool getwindowname $j`
        case $title in
            **)
                echo "Found running Gedit with unsaved data"
            ;;
            *)
                echo "Gedit is open but there are no unsaved documents"
            ;;
        esac
    done
done

What this script does. It iterates over all found gedit processes, finds their visible windows, iterates over their titles and checks if the title contains * as the first character which indicates an unsaved document.

Bash/shell purists will probably want to replace `` with $( ).