Does a religion have to be the one displayed in the city banner for it to apply? I have seen some cities with two or more religion where no city banner icon popper up.
In most cases, only the majority religion (the one displayed on the banner) gives any benefits to the city. There are a couple of exceptions, such as the founder belief "Tithe" which is granted for every 4 followers, regardless of the city's majority religion.
I think I get followers as simply just how many are following the religion (well, simple enough), but am unsure how the pressure is determined / defined / worked. I have seen on several occasions where majority was following A religion while B religion had more pressure.
Pressure is a set number (which varies by game length), multiplied by the number of other cities within 10 tiles which have followers of that religion. For example, in a standard length game each city with a religion exerts a pressure of 6 on all other cities within that radius. Both the radius and the pressure/city can be raised with enhancement beliefs (Itinerant Preachers and Religious Texts, respectively). In simple terms, if City A has 5 other cities within 10 tiles in which at least one person is following Christianity, then Christianity has a pressure of 30 in City A. The holy city might exert slightly more pressure, I'm not certain. If you see that a minority religion has a higher pressure in a city, that's basically saying that over enough time and without any outside interference (missionaries, great prophets, or inquisitors used on that city or others within its pressure radius) that religion will become the majority.
Does great prophet just outright delete other religion off the city? I have had my holy city (and I have done the same to other civ as well) become unholy(...) / converted to other religion with my religion gone from the city. Do I gain the holy city status back, when I reinstate my religion back in the capital?
As far as I've seen, yes. The great prophet sort of acts like an inquisitor and a missionary all rolled into one, and also has the advantage of not suffering attrition damage when inside foreign territory without a treaty. I've never personally seen a holy city stripped of its status (but have seen other people mention it on here), so I don't honestly know if you would get that back by re-inserting your religion into it.
In a game where your neighbors are aggressively converting your cities and you don't have the military muscle to make them stop (with extreme prejudice

), it's good practice to park a friendly inquisitor in your capital & other cities where maintaining your religion is crucial. Missionaries and great prophets cannot convert your cities when an inquisitor is there.