Actually doesn't necessarily need to be a child of his by Mary. While Anne would still be the heir of William and Mary, if William had a child by a later marriage after Mary's death this child would become Anne's heir if she doesn't have surviving children as ITTL. So you still have a possibility of union with the Netherlands even without Mary having children.
But this requires finding another bride for William, his fathering children, and all of Anne's dying as per otl. While this does raise the possibility of a union with the Netherlands, it seems to me that, in general, it is easier to get a personal union through one of Anne's children.
The Dutch (actually the province of Holland and a couple of others) got rid of th stadholder twice. Both because the succession wasn't clear. Once when Willem II died and his son Willem/William III wasn't born (or just born) and once when Willem III died and left no sons. So I think that if William III left a clear successor, Holland (with a little English pressur) would choose him as the next stadholder. Although I must admit that he would probably the last English stadholder, certainly if there is another clear choice (either he has two sons, one becoming the king of England the other stadholder of Holland etc. or they just choose the Frisian stadholder like OTL).
Agreed. With one son surviving to adulthood (or just outliving the father), the dutch are more likely to keep the office of stadholder, I'm not so sure if they would jump at the prospect of a continued union. To certain parties this would signify the loss of too much independence...
BTW about children of William III but not from Mary. I thought that William had his own claim to the English throne and was an actual king of England. Why exactly wouldn't his children succeed him as king? I would understand it if Anne became queen after Mary died, but she only became queen after William died. Could someone explain that to me?
William does have a claim on the throne, but it is secondary to Anne's. Anne is the younger daughter of James II; William;s claim comes from his mother being a daughter of Charles I. As for the issue of rule and succession, it all goes back to the aftermath of the glorious revolution. There was some dispute over who, and in what capacity, should succeed to the throne, so in the end, after a row with the Lords, William and Mary were crowned co-rulers. However, since William's most direct claim on the throne was as the consort of Mary, Mary's children also take precendence. This is actually specifically laid out in the Bill of Rights; Mary's heirs have the first claim on the throne, followed by her sister Anne and her children, and then to William and his children not also children of Mary, as the only child of Mary, Princess Royal (Charles I's eldest daughter).