Does completing an urgent task requested by the manager before performing the prayer—knowing that the prayer will not go out of its time—constitute shirk or incur sin, and is an employee's prayer in his office instead of the mosque, due to the imam's prolonged "takbirat al-ihram," correct or sinful?

As long as you do not delay the prayer beyond its prescribed time, there is no sin upon you for performing it after work. Prolonging the hamza of the word 'Allah' in the opening takbir (Takbirat al-Ihram (definition)) is a mistake (lahn) that invalidates the prayer according to some jurists, because it turns it into a question. The imam should be advised, and if he does not correct his mistake, he must be replaced. If that is not possible, it is permissible for you to refrain from praying in his congregation and instead pray in congregation with other employees.