Should a manager use written formal communication method for an employee's non-performance at once?

Yes, I agree. However, the written communication should be in a very positive language with an action plan on how to improve the performance.

Well, coaching will be a first method, it will also involve a formal written communication. Doing a one-on-one closed door discussion with your non-performing staff is the appropriate step, but what was not written was not done, therefore even if you do a coaching method, you need to document it and it will serve as a formal communication between you and the employee. 

Yes and No. The degree of the degree and number of times of the non-performance will dictate the need for written formal or oral communications.

In high degree of non performance and several non performance written formal communications could be given at once but for minor non performance and first time non performance oral communication could do.

In what ever circustance, the communication should competence improvement driven rather than a source of discouragement.