Once the screen is well calibrated you can assess it's capabilities -
I feel that measured tests can only go so far - especially since measurement normally uses the same instrument used for initial calibration.
I have seen instruments with internal faults return awful calibrations - and then good verifications.
In my opinion, a visual comparison to a printed reference [properly illuminated] is the only way to asses display capability properly. What "looks good" in a users opinion is of little value, I'm sure you realised this and that's why you're asking.
There is an RGB and CMYK test image on my own website [http://www.colourmanagement.net/downloads.html] that's freely available for you to download and open in Photoshop assess colour and tonal reproduction - even without the reference proof it has some use since "memory colours" e.g. fleshtone are present, as is monochrome.
Resist the temptation to assess the appearance against a tear sheet or print you've made / had made. You need a certified proof to be sure that what you're using as reference is "actually right".
You mention uniformity, a measure of the screen's capability to display a tone or colour evenly all across it's surface.
Again, testing that is mainly a visual process, to get started just put up a neutral grey without any icons. Does it look even - or at least within reasonable tolerances?
IF the uniformity of the screen is outside what you consider reasonable then juts go back to the manufacturer. They will have tolerances for this. Sadly pretty high deviations are considered to be "within tolerance" in many cases, especially as display price drops.
I hope this helps
enjoy the new screen