If Bil'am was a prophet (who saw angels whenever they prophesied), why did Hashem need to open his eyes to see an angel?
Ramban: This Pasuk is indeed a proof that he was not a prophet, 1 but a plain sorcerer 2 - who initially received the word of Hashem in that capacity, 3 and who will shortly merit a 'vision of Shakai'. 4
Ramban: And it was people who were not prophets, such as Hagar (in Bereishis 21:19) and the servant of Elisha (in Melachim 2, 6:13) who needed their eyes to be opened to see angels.
Ramban: As the Pasuk describes him in Yehoshua, 13:22.
Ramban: See above, Pasuk 8.
Later in 24:16, in honor of Yisrael. See Ramban, end of Dibur.
Why did Hashem show Bil'am the angel with his sword drawn?
Oznayim la'Torah #1: "With words alone an Eved will not be chastised", 1 and it was only after he saw the Mal'ach with his sword drawn that he conceded "Chatasi".
Oznayim la'Torah #2: In order to justify the actions of the ass in his eyes, to admit that he had smitten it unjustly, and to acknowledge that the ass had seen what he had not seen and that it had saved his life.
To warn him that his current actions did not find favor in the Eyes of Hashem and that he should retract.
