Now since all the three companies are sister companies and as one of them has shut down and you all are transferring your employees from the closed firm to another firm, there should be gratuity angle considered. You must be having employees who might have served more than 5 years or are about to finish the said continuous tenure.
Acquisitions and Mergers are not just the handling issues for finances and employees, but HR have to be on their toes to see to it that employees are not hampered.
The PFs should be transferred as it would have and change of company's name would not have an impact on it.
You will have to issue new appointment letters to the employees in the new company's name. However do mention on the new appointment letter that the prior experience with their ex-company named [the shut down company's name] would be considered for gratuity purpose.
Waiting for seniors to guide better.
for ESIC it will be better to open a new account as pending contribution may creat hurdle for you.
as far as gratuity is concern you have to place a request to merge gratuity fund for both the companies.