I want to delete my current free account and create a new one using a different subdomain. Can I register again using the same contact details I'm currently using right now?
Yes, but not until the system has cleaned up your existing account, which can take some time. (Since you can't have direct admin access to the server, your action basically just marks your account for removal. Scheduled actions take it from there.) You'd sign into your portal account (at https://x10hosting.com/sso/login/ - which won't be deleted) and create the new hosting account from there.