I have both an NX1000 and NX3000, and have both lenses. I’m a casual user and can’t provide a detailed critique. I bought the NX3000 to get the 16-50mmPZ at an attractive price.
I wanted the 16-50mmPZ because:
1 — I wanted to be able to shoot at 16mm, which is significantly wider than the 20mm and really makes a difference in landscape shots and group shots
2 — I wanted the OIS capability, primarily for indoor available-light shooting
3 — I wanted a front element that did not rotate and move in and out with focusing so that I would not have to touch up the rotation of a CPL filter when I have one mounted. The front element on the 20-50mm moves and rotates with focusing, and I would get lens errors sometimes when adjusting it after focus, whereas the 16-50mmPZ has no external change. The 16-50mmPZ front does move in and out with zooming, but I don’t find that a problem.
The 16-50mmPZ also has the power zoom feature through buttons on the lens barrel, but I find myself doing it manually anyway (the zoom ring just drives the same motor anyway; it is not mechanically linked as far as I can tell).
I have found the 16-50mmPZ to be a great lens, wonderfully compact for its capabilities, and for all of the above reasons, it has become the lens I most use on the NX3000. The NX1000 is mostly used for legacy lenses via an adapter, but is also a very capable back up.
To address your other 2 questions, I have used the 16-50mmPZ on the NX1000, and it works very well. The NX3000 has several different features. Foremost in my mind is the flip-up LCD (much more useful than I thought it would be). Others have commented that the jpeg engine in the NX3000 is improved, and I don’t doubt that, but I have been pleased with the jpegs out of both cameras in that regard. There are other minor differences on the NX3000, such as a micro-SD card (which I find annoying, and seemingly unnecessary), a different cell phone-type battery and no external charger (I bought a spare battery with a compact charger of ebay for $20 - same as used for a Samsung Galaxy S4 zoom phone - to permit external charging), a less-pronounced bulge on the grip, and a different on-off button. The software displays are also formatted somewhat differently, with the NX3000 foregoing the multiple screens within a given category for scroll-down lists.