How to get the current monitor resolution or monitor name (LVDS, VGA1, etc)

You should be able to do this by a combination of xrandr and xwininfo.

1. Get the screens, their resolutions and offsets:

   $ xrandr | grep -w connected  | awk -F'[ \+]' '{print $1,$3,$4}'
   VGA-0 1440x900 1600
   DP-3 1600x900 0
   

2. Get the position of the current window

   $ xwininfo -id $(xdotool getactivewindow) | grep Absolute
    Absolute upper-left X:  1927
    Absolute upper-left Y:  70
   

So, by combining the two you should be able to get the resolution of the current screen:

#!/usr/bin/env bash

## Get screen info
screen1=($(xrandr | grep -w connected  | awk -F'[ +]' '{print $1,$3,$4}' | 
    head -n 1))
screen2=($(xrandr | grep -w connected  | awk -F'[ +]' '{print $1,$3,$4}' | 
    tail -n 1))

## Figure out which screen is to the right of which
if [ ${screen1[2]} -eq 0  ]
then
    right=(${screen2[@]});
    left=(${screen1[@]});
else
    right=(${screen1[@]});
    left=(${screen2[@]});

fi

## Get window position
pos=$(xwininfo -id $(xdotool getactivewindow) | grep "Absolute upper-left X" | 
      awk '{print $NF}')

## Which screen is this window displayed in? If $pos
## is greater than the offset of the rightmost screen,
## then the window is on the right hand one
if [ "$pos" -gt "${right[2]}" ]
then
    echo "${right[0]} : ${right[1]}"    
else
    echo "${left[0]} : ${left[1]}"    
fi

The script will print the name and resolution of the current screen.