As to the wights:
1) There is no "infection" or rule that they must be killed by a WW to turn into a wight. There is also no rule that "any dead person north of the wall automatically turns into a wight". We saw the NK raise the wights at Hardhome with his hands. It's certainly possible (and I would argue probable) that the other WWs can also raise them within a certain distance. There's certainly more magic involved in making a WW than a wight. We saw the wights rise instantaneously, but Benjen said the Night King was trying to turn him into a WW when the CotF saved him, leaving him some sort of half-undead magical being.
2) That means the wight that came under the wall was reanimated beforehand. We KNOW this because Jon noticed it didn't smell and wasn't rotting meaning it had already been reanimated. Therefore, it was a Trojan horse move by one of the WW (if not the NK himself). Perhaps he was testing the wall to see if the magic had worn off? Perhaps he was checking to see if there were any rules around "the wall stops magic, but magic starts working again on the other side of the wall". Either way, it was clearly a Trojan horse. It is also possible that wights can go south of the wall, but the WWs can't so that's to what Benjen was referring when he said he couldn't go south of the wall.
3) All the remaining freefolk were congregated at Hardhome. Obviously the WWs would attack it as soon as they realized that's where they were congregating. The only reason anyone escaped, as someone pointed out earlier, was because Jon showed up with a bunch of boats at the last second.
4) I don't want to make too much about this since I've said numerous times that the WWs are waiting for the "right" time to attack the wall, but the OTHER difference between Hardhome and Eastwatch is that one was a big refugee camp FULL of defenseless people and a mostly useless wooden gate. Eastwatch is only manned by a small army of trained warriors and significantly better defenses (not counting any magic still related to the wall). It is clear that the NK is utilizing strategy and identified Hardhome as an easy target filled with tons of new bodies. Eastwatch is simply an obstacle to get past when trying to get to Winterfell.
Hopefully that clears things up for some people.