Registrars job is primarily to coordinate and make sure there are no duplicates in domain names. ICANN manages them. At a technical level registrars (and registries) use the Extensible Provisioning Protocol to achieve this.
They do update the DNS with nameserver information but not the Root DNS servers (which is an entirely different area) only the TLD (Top Level Domain) servers.
Legally (I am not sure about this, so take it with a pinch of salt) you have a contractual relationship with your registrar and registry, and from there you can take it to courts and so on. They are governed by the country they reside. Registries also have pretty comprehensive dispute policies in case of disagreements of the ownership for example. However, starting with ICANN, all these organisations (another example is Nominet in the UK) have massively policy based working style, and they update them according to the feedback mainly from their members and customers (us).