How can you rotate the sprite and shoot the bullets towards the mouse position?

There are 2 issues. The current position of the bullets is multiplied by its the normalized direction vector. That doesn’t make sense:

for bullet in list_of_bullets:
   bullet[0] = bullet[0] * bullet[2][0]
   bullet[1] = bullet[1] * bullet[2][1]

Add the direction vector to the position of the bullet:

for bullet in list_of_bullets:
    bullet[0] += bullet[2][0]
    bullet[1] += bullet[2][1]

If you want to increase the speed of the bullets, then you’ve to scale the vector by a certain speed. e.g.:

def spawn_bullet(x,y):
    global list_of_bullets
    initial_x = x
    initial_y = y
    mouse_x, mouse_y = pygame.mouse.get_pos()
    vector_x, vector_y = mouse_x - x, mouse_y - y
    
    distance = math.sqrt(vector_x ** 2 + vector_y **2)
    speed = 5
    move_vec = (speed*vector_x/distance, speed*vector_y/distance)
    
    list_of_bullets.append([initial_x, initial_y, move_vec]) 

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The 2nd issue is, that the display is cleared, after the bullets are draw, so you’ll never “see” the bullets.

Draw the bullets in update_game after the display is cleared:

def update_game(x,y,width,height,char):
    win.fill((0,0,0))
    for bullet in list_of_bullets:
        pygame.draw.rect(win,(0,0,255),(int(bullet[0]),int(bullet[1]),20,20))
    point_to_mouse(x,y,char)
    pygame.display.update()

(Delete the drawing for the main application loop)

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Minimal example:

import math
import pygame

def blit_point_to_mouse(target_surf, char_surf, x, y):
    mouse_x, mouse_y = pygame.mouse.get_pos()
    vector_x, vector_y = mouse_x - x, mouse_y - y
    angle = (180 / math.pi) * -math.atan2(vector_y, vector_x) - 90
    rotated_surface = pygame.transform.rotate(char_surf, round(angle))
    rotated_surface_location = rotated_surface.get_rect(center = (x, y))
    target_surf.blit(rotated_surface, rotated_surface_location)

def spawn_bullet(list_of_bullets, x, y):
    mouse_x, mouse_y = pygame.mouse.get_pos()
    vector_x, vector_y = mouse_x - x, mouse_y - y
    distance = math.hypot(vector_x, vector_y)
    if distance == 0:
        return
    speed = 5
    move_vec = (speed * vector_x / distance, speed * vector_y / distance)
    list_of_bullets.append([x, y, move_vec])

pygame.init()
window = pygame.display.set_mode((500,500))
clock = pygame.time.Clock()

rocket = pygame.image.load('Rocket64.png')
rocket_rect = rocket.get_rect(center = window.get_rect().center)
velocity = 6
list_of_bullets = []
bullet = pygame.Surface((20, 20), pygame.SRCALPHA)
pygame.draw.circle(bullet, (64, 64, 64), (10, 10), 10)
pygame.draw.circle(bullet, (96, 96, 96), (10, 10), 9)
pygame.draw.circle(bullet, (128, 128, 128), (9, 9), 7)
pygame.draw.circle(bullet, (160, 160, 160), (8, 8), 5)
pygame.draw.circle(bullet, (192, 192, 192), (7, 7), 3)
pygame.draw.circle(bullet, (224, 224, 224), (6, 6), 1)

run = True
while run:
    clock.tick(60)
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            run = False
        if event.type == pygame.MOUSEBUTTONDOWN:
            spawn_bullet(list_of_bullets, *rocket_rect.center)

    for bullet_pos in list_of_bullets:
        bullet_pos[0] += bullet_pos[2][0]
        bullet_pos[1] += bullet_pos[2][1]
        if not (0 <= bullet_pos[0] < window.get_width() and 0 < bullet_pos[1] < window.get_height()):
            del list_of_bullets[list_of_bullets.index(bullet_pos)]
            continue

    keys = pygame.key.get_pressed()
    rocket_rect.x += (keys[pygame.K_RIGHT] - keys[pygame.K_LEFT]) * velocity
    rocket_rect.y += (keys[pygame.K_UP] - keys[pygame.K_DOWN]) * velocity
    rocket_rect.clamp_ip(window.get_rect())
    
    window.fill(0)
    for bullet_pos in list_of_bullets:
        window.blit(bullet, bullet.get_rect(center = (round(bullet_pos[0]),round(bullet_pos[1]))))
    blit_point_to_mouse(window, rocket, *rocket_rect.center)
    pygame.display.flip()

pygame.quit()
exit()