Would she have carried out that threat on a popular prophet?
Definitely yes. Jezebel was an evil person.
1 Kings 19:1 Now Ahab told Jezebel everything Elijah had done and how he had killed all the prophets with the sword. 2So Jezebel sent a messenger to Elijah to say, “May the gods deal with me, be it ever so severely, if by this time tomorrow I do not make your life like that of one of them.”
Ahab killed a whole bunch of prophets already. What's one more? The threat was so real that
3 Elijah was afraid a and ran for his life.
Later Jezebel plotted to kill an innocent man Naboth in 1 Kings 21.
Why go to the furthest place possible (Mt Horeb) Why not go to Judah where he was from to be protected by the King of Judah?
At that time, Elizah did not trust the King of Judah. Israel and Judah had a good diplomatic relationship.
Later, in 1 Kings 22
2 But in the third year Jehoshaphat king of Judah went down to see the king of Israel.
3The king of Israel had said to his officials, “Don’t you know that Ramoth Gilead belongs to us and yet we are doing nothing to retake it from the king of Aram?”
4So he asked Jehoshaphat, “Will you go with me to fight against Ramoth Gilead?”
Jehoshaphat replied to the king of Israel, “I am as you are, my people as your people, my horses as your horses.”