Dear Yogyata,
You may use the letterhead with the registered office address of the USA.
As the company has not started its operations in India, nor do they have an office in India, it would be better that you do not employ people as employees but as Consultants. A letter stating their consultant status with the company, their compensation and benefits, and also their roles and responsibilities to the respective people.
The company may also enter into a MoU with the so-called consultants, assuring them that they will be absorbed by the company as soon as the company fulfills the required statutory provisions (only if the need arises, the MoU needs to be worked out between your Boss's company and the employee).
As a matter of precaution, the jurisdiction in case of any issue arising can be kept as both India and the USA, with the primary cause for any legal case of action arising in India.
By following the above procedure, the company can temporarily evade company law complications. Also, to a certain extent, it would be easier for your boss to wind up the business if the same doesn't do well, as all the employees employed with the company would be consultants.
This procedure is applicable only for top and middle-level management cadre people and not for staff. If something similar has to be worked out for staff, then a different approach has to be adopted.
Hope this solves your query.
Thank you,
Octavious