I had a great William III dream, but have a few questions before I would attempt to make anything of it
I see that his cousin William IV became Stadtholder in the Netherlands - is this because William III was childless, or was it basically elective and they chose him as the nearest relative?
The thing you need to realise about the stadholdership is that it is not the same thing as a monarchy. For example, in the time of William III there was no stadholder of the Netherlands. There were only stadholders of the seven provinces (and Drenthe). William III for example was only stadholder of Holland, Zeeland, Utrecht, Gelderland, Overijssel and Drenthe. Someone else was stadholder of the remaining two provinces Friesland and Groningen. Still, because William III was stadholder of the most important province (Holland) he was the most important stadholder in the Netherlands. Stadholders were basicly appointed by the provinces and in theory everyone could become stadholder, but it became basicly tradition to appoint the nearest relative of the former stadholder, like William II who became stadholder after his father stadholder Frederik Hendrik died.
To complicate it though, you have to remember that many of the people who ruled Holland (and some other provinces), the regents (usualy rich merchants), prefered not to have a stadholder. Even though the stadholder was technically not powerful, he usualy was very influential and the regents prefered to rule themselves. So when Willem II (the father of William III) died, when his only son was still very young, the regents decided they did not need a stadholder. So William III had not been stadholder for the first couple of years of his life. Only when the regents screwed up (at least in the eyes of the people) during the year of disaster, when the Netherlands was attacked by France, England, Cologne and Münster, William III was appointed stadholder, hoping he could safe the country (he was).
So when William III died without children there was no clear successor to him. William III appointed his (relatively distant) relative, the stadholder of Friesland and Groningen, as his successor, but the regents prefered not to have a stadholder. So there was a second stadholderless era, until during a war the people wanted William IV, current stadholder of Friesland and Groningen as stadholder of Holland, Zeeland, etc. Only from that point the stadholdership was hereditary and the stadholder was stadholder of all provinces.
BTW The Frisian stadholder was not the closest relative of William III, that would have been the ruler of Brandenburg/Prussia.
If William III had a son, would they have still have looked to have separated the crowns?
It depends on the age of the son. Personaly, my guess would be that the "personal union" (which was not an actual personal union) would have continued, at least for a while. I think at one point the regents would have enough of the English kings and appoint someone else as stadholder. Although, if the king of England spends his time in England and leaves the Netherlands alone, they might be happy anyway.
Luxembourg at this time appears to have been somewhat in flux, could Britain or the Netherlands have claimed or bought it from the Habsburgs?
As far as I know Luxemburg was just part of the Southern Netherlands, not that different from Flanders or Namur. I see no reason why the Netherlands or Britain could claim it. They could buy it of course, but why? There are better parts of the Souther Netherlands to buy.