having discovered from prophecy
I admit this was ambiguous. The acount tells us that the chief priests and scribes worked it out from prophecy, they informed Herod who told the Magi. I omitted those intervening steps, not because I didn't think they were in the account as they clearly are, but just for brevity.
It was logical for them to go to Jerusalem as that was the capital where the ruler would live: Bethlehem was just a village near Jerusalem.
Second, it doesn't say that the star appeared, then disappeared,
The Magi saw the star in the east (v10). This is aorist tense: a single completed action in the past.
The Magi saw the star again on the way to Bethlehem. This is aorist tense: a single completed action in the past.
The star then led the magi to Bethlehem. This is imperfect tense - the star leads them to Bethlehem and to Jesus: only now do we have an ongoing leading.
Given that Herod had all the children up to the age of two killed, it's likely that the star first appeared to the Magi two years before they came to Jerusalem. (Herod asked them when
the star had first appeared - why do you think he did that?)
So yes, it was visible all the time, probably hanging above Jerusalem
The text and the verb tenses do not support this view
What matters is that the star led them to Jerusalem first
The text does not say this.
I don't know what you are trying to say here.
Num 24:17 may be Messianic although Luther doubted it with the source being Balaam. Yes the star here is metaphorical of authority but there is an allusive connection between Num 24:17 and the Star of Bethlehem.