For the first question: the resolving power of the average human is around 20 arcseconds, which means that individual pixels cannot be resolved once they appear smaller than about 0.00009 radians (say, horizontally). So assuming your screen takes approximately 0.5 radians of your field of view, which is what you get for a normal monitor viewed at a normal distance, you would need a resolution of at least 5550 pixels horizontally (multimonitor setups don't count, since they are supposed to span a larger angle in your field of view).
Given that your average 22" 1080p monitor (1920 pixels horizontally) has a PPI of 96 pixels per inch, you'd only need around 3 times that, that is, a PPI of around 300, beyond which a typical human cannot distinguish individual pixels (this appears to be more or less confirmed in this answer).
I don't know for the second question. Perhaps someone else would care to answer it in detail, but I imagine you would still need maybe double, triple that for aliasing to really disappear, as at 300 PPI you are still at the limit, where nearby pixels aren't quite yet averaged together by the sensor (i.e. human eye), they are just barely distinguishable. But that's just a guess.