I've seen threads where people talk about certain problems like dead/stuck pixels on new monitors that arrives that they check for so they can return it immediately within the return window if it has these problems. I'm just wondering if there are any other similar problems I should check for when my monitor first arrives, and how I should go about doing this.
The period for return is usually the warranty period that you've paid for when buying the monitor. In that period if you see dead pixels or stuck pixels, you can contact seller and then initiate an RMA. Do keep in mind that seeing just one pixel that's dead or stuck does not qualify for an RMA, it needs to be a significant amount, more along the lines of 7 or more. Your product guide/manual should state that(in Asus's manual, it's listed on how many pixels need to be dead before the unit qualifies for an RMA).
Assuming that the monitor appears to work ok, you really need to do nothing.
If you want, there is a dead pixel tester: http://lcdtech.info/en/tests/dead.pixel.htm
Some monitors are sold as "perfect pixel"
For them, any dead pixel found is a reason for return.
Most of the time you will not notice a few dead spots out of millions.
If they are in the center, then perhaps.
The particular monitor you buy should have warranty rules.