First question: Yes, William (not Mary) maintained his position as Stadtholder of the Dutch Republic. It's worth nothing that though officially dual monarchs of England, in practice William took the leading role even while Mary was still alive.

Second question: You can quibble about this, but I'd say in principle, no.

For this, you need to understand that England and the Netherlands under William III were not absolute monarchies.

The Netherlands was a Republic and William's position was more comparable to that of a parliamentary president and commander-in-chief for life as opposed to a traditional monarch. I.e. his position was that of a servant of the state, and his power was heavily circumscribed by the rights and politics of the provinces and cities as represented in the States General. In particular the patrician government of the city of Amsterdam with its massive wealth and thus considerable influence on the national budget, was a constant power factor for William III to manage. One that often went directly against the policies desired by William.

In England the power balance between the monarch and parliament had been a continual issue for at least a century, leading to a history of armed conflict of which William's invasion was the latest chapter. William's success in achieving a largely bloodless victory (in England, the situations was quite different in Scotland and Ireland) lay in part in William's strategy of applying the lessons he'd learned in the Dutch Republic to managing his relationship with the English parliament. William, in principle, was experienced with and prepared to accept a more subservient position to parliament than many of his homegrown predecessors could stomach. This was aided by the fact that William's invasion of 1688 had been largely motivated by foreign policy, England's position in the continental balance of power, in which William saw England as a crucial 'swing vote' determining the Dutch Republic (and the whole of Europe's) security or insecurity versus the power of Louis XIV's France.

Thus, under William III, the foreign policy of England and the Dutch Republic became largely the same, i.e. anti-French, and in that sense you could argue that they were sort-of the same country.

But I think that the question implies a much stronger union, which there never was. Though there was a considerable exchange of knowledge and people (quite a few Dutch aristocrats and merchants used the opportunity to immigrate and become English aristocrats and merchants), with the exception of William himself the governments of both countries continued to function largely autonomously. There was no merger of the Dutch States General and the English parliament, domestic policy remained entirely local to both, and foreign policy was only harmonized in so far as William managed to get his way in both the States General and parliament (which was never assured and rarely easy.)

This is also why when William died the union dissolved pretty much immediately. Its legacy being primarily England's geopolitical foreign policy, which was continued largely along the course William had set it on, with England remaining an ally to the Republic and a counterweight to French hegemony essentially up till the French threat was replaced by the German threat (barring, of course, noted low-points in Anglo-Dutch relations such as the fourth Anglo-Dutch war.)

Someone else might have another take on any specific detail, but I think this is the tl;dr of what you were asking about.