How does domain registration work?

1) The registrar has a DNS server. It tells the main (called top-level) DNS server owners (for instance the .com) to forward all requests for yourdomain.com to their DNS server. You can also do this directly with the owners of the top level domain (ICANN in the case of dot com I think). But then you need to run your own DNS server.

2) They don't, see number one.

3) They pay a fee to the top level registrar.

4) Depends on your registrar (possibly country), but with proper registrars you do (however read the contract with your registrar).

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).