This made me curious as to what makes a question on this network "hot".
There is a certain score for that. Note that the question and answer(s) first must have some upvotes from the site itself before the question can become a Hot Network Question. After the hot network period ends, it's usually a matter of how often people Google for the problem. Somehow this Puzzling question of mine consistently gets about a hundred views a week.
Do certain tags improve the visibility?
That depends on the site. On large sites, definitely; on Stack Overflow, most people watch one or more tags and usually completely ignore the other questions. On smaller sites, you have more people who read almost everything, and an attractive (possibly even 'clickbaity') title is more important than tags.
Does posting at a certain time matter?
I doubt it. I've conducted some research a while ago about posting times and concluded they had no influence (the reduced traffic is compensated by the reduced number of questions vying for that traffic). But it could be that posting during the weekend or outside European and American office hours will make it hard for a question to compete with the ones already in the list.
Yes, it does; it's best to post them on Sunday. Here is a SEDE query which lists the % of questions which become a Hot Network Question. The x-axis starts at Sunday 0:00 UTC, and you see that after 30 hours, when the workweek begins, the percentage drops significantly, only rising again when it's Saturday.
I get similar results when excluding Stack Overflow (which makes up for most of the questions in the network), but even when limiting to a single site, e.g. Worldbuilding (using 8-hour buckets for clarity), the effect is clearly visible:
Why would this be the case? There is less voting during the weekend, and I suspect 'hotness scores' are lower (I should be able to check that, since I regularly scrape the list) so it's easier for new questions to enter the list.