This is a very interesting question which I looked up recently. The most comprehensive treatment I found is by R Doniel Neustadt, author of the Daily/Monthly Halacha books. This is found in the Ohr Olam Mishna Brura vol 3a p. 441. See there for detailed sources (or ask in the comments)
R Neustadt doesn't formally rule on the topic but explains it as follows.
The Torah obligation to keep Shabbat doesn't apply to one's machines or tools which is why we use lights and air conditioners. Nevertheless he sees the following possible three Rabbinic restrictions
1. Mekach u'mekar: the prohibition of any kinyan (acquisition) on Shabbat (even without writing contracts or exchanging money) to avoid possible Shabbat violations such as writing. On the other hand, since an online transaction is completely automated without any active involvement or knowledge from the site owner, he says this doesn't apply, in the same way one can give money to a non-Jew to buy an object on Shabbat if we do not instruct him to do it specifically on Shabbat and he can do it at other times as well. He therefore concludes "passive business" is allowed on Shabbat
2. S'char Shabbat: the Rabbis prohibited profiting from a forbidden activity engaged in on Shabbat. But the poskim understand this to be related to services rendered on Shabbat, not to sales (e.g., food bought on credit on Shabbat is allowed). Thus a website owner if permitted to keep his profit from Shabbat shales
3. Zilzul Shabbat: the remaining intangible but crucial issue is the one of desecration of the sanctity of Shabbat since business will be conducted seven days a week without regard for Shabbat or Yom Tov. It might be a violation of the spirit of Shabbat. He doesn't rule on this one and calls for the leading poskim of the generation to weigh in
As you saw in the other question you linked to, there are a number of lenient opinions, but we also see that leading Jewish e-commerce sites (e.g., artscroll) are closed on Shabbat.
It appears to me based on the above that there is room to be lenient on your first question (since no business transaction occurs). But you definitely need to CYLOR.