if you look at the pictures for the Asus monitor on newegg.com link you provided. there is an input source selection button right next to the power button. press that and you can switch between the laptop and xbox.
why wouldn't the xbox be able to run at the same time as your 15 inch monitor?
you can have your xbox, car, tv, gps system, smart phone, hvac, water heater and stove running at the same time.
which monitor? what do you mean compatible WITH the asus monitor?
most monitors don't connect to each other, they connect to the computer. separately.