Focus on the two Registeels.
It proves
(a) The GO symbol is tied to Go origin game and Go Met Location. (LGPE mons from GO doesn't have it)
(b) It implies that "barred entry into a game" could be a byte, or some other form of data on the pk8. If it wasn't, either both Registeel will be disallowed entry, or both Registeel will be allowed entry. (The cross symbol)
The user chanced upon the oddity when testing my claim of Go mons needing to be Go origin game and Go met location.
(The user was using a Zapdos, and was puzzled by why it appeared to allow entry when that SWSH save doesn't have a Zapdos).
I got the user to have both mons side by side in one image, to confirm the situation.
(As I said, if the HOME checking wasn't dependent on some other value, either both Registeel will be disallowed entry, or both Registeel will be allowed entry. Not half and half.)
(1) HOME ID is re-assigned whenever it re-enters HOME. The mon will persist not being able to enter SWSH saves that never had that species. I tested that for myself.
SW caught Melmetal, SH didn't have it. GO Melmetal couldn't enter SH, despite being in SW and re-entering SH.
You could argue that they could just recalculate the HOME ID to match that formula on re-entry, but thus far the HOME IDs just climb sequentially, doesn't appear to adhere to any weird jumps to match a formula.
(2) As per what I said in number 1, it "does not get cleared". The requirement to enter a save that has had it persists.
In any case, I'll like to attempt more tests to narrow down the phenomenon.
Also, apparently only mythicals/legendaries has that entry requirement. Since we know HOME isn't checking by species (the Registeel example above), it'll make sense that they leave a bit/byte on it somewhere, then just keep checking if the bit is flipped. My next order of business, would be to check the fateful encounter byte (I think it's shared with other stuff), could be fun.