Can i claim gratuity?

You can make a complaint with the grauity commissioner about this or can file a suit for recovery of your gratuity amount which can be calculated by you as per your total employment tenure. 

Yes you seem  to have a good case (atleast appearing from your descripttion).

Yes you can legally claim gratuity benefits in your case? 

Write a letter to your company HR department and sign and send scan copy by email and simaltenously send letter by Speed post and ask for your gratuity claim for 10 years or more. If the company denies your claim then complain online https://pgportal.gov.in/ in this website 

an employee when gets transferred within one of the group company, the understanding and arrangements within such Group Company has to be in place so that such benefits are structured in alignment with the transfers.

When the 4 Labour Codes becomes effective, as per the new gratuity rules under the Social Security Code, the employees will be enabled to transfer the gratuity benefits as well like that of the EPF benefits.

Gratuity cannot be calculated to carriy forward, it has to calculated for the period served with the organisation in terms of Payment of Gratuity Act.

It is change of name of the company did not change the employer / management, which is just to distract the authorities and public in general. Accordingly your service is continueous with the employer hence the managagment is liable to calculate and pay gratuity for entire period, claim it.

What does company has changed 3 times means?

The name of the company changed? Was the A company takenover by the B company and thereafter by C company? Since one of your appointment letters stipulates that your gratuity will be continued, it can be assumed that one company took over the other and you continued working without any break. If that is the case, then obviously you can claim gratuity against the last employer for entire period of service. 

Demand gratuity by sending FORM I to the last employer and wait for 30 days.

If gratuity is not paid within 30 days, file an application before the Controlling Authority (mostly labour commissioner) in FORM N and demand outstanding gratuity and interest.