The email quoted does not say anything about domain ownership. It says that if you use the hosting hosting company's tools to modify the contents of the web site, you might cause problems, and if that happens, the web design company Company_Name will not be responsible. Further it says that if that happens, and you want Company_Name to fix the problem, there will be an additional charge.
As to ownership, the web design firm will own the copyright on the web code that they wrote, unless your contract specifies that you will own it from the start (work made for hire) or that they will transfer it to you (assignment), perhaps after you pay for it. If you supplied any documents for them to place on the site, the copyright to those will remain unchanged, Company_Name will not get ownership of such copyrights.
(Note that, in the US at least, "work made for hire" can apply to a contract job, but only if the contract explicitly says so. I don't know if that is true under the law of India.)
As to ownership of the domain, that will depend entirely on the language in the contract that you signed with Company_Name. If you paid for the domain registration, it is reasonable for you to own the domain, but the contract will control.
The quoted email implies that you have been given the passwords to the cpanel and ftp accesses to the site, which would allow you to change anything on the site, or engage a different firm to make future changes. This would typically imply that you had "ownership" of the site, but again, everything depends on the wording of your contract with Company_Name.
Note that ownership of the copyright to the web code used, ownership of the copyright to other content displayed on the web, ownership of the domain, and ownership of the site (i.e. the right to make and control changes to it), are all separate, and it would be possible for each of these to rest in a separate entity. It would be common for the design firm to retain ownership of the web code, and license it to the client. ownership of the domain and the site would be more likely to go to the client, but the design firm could retain all or any of these, if that was the agreement. Any not owned by the site owner would need to be licensed so that the site owner could use them. Such a license might be permanent, or might expire after some period.