Correct for new domains; see below.
This will not apply to domains registered under our older TLDs (com/net/org/info), only to domains registered under new TLDs. I can't rule out the possibility that we may partner with other registrars in the future for new TLDs, but for the moment I believe that Directi will be the only new name you see.
This depends on which TLD you're registering a domain under. Not all registries allow private registrations — for instance, .US requires that the registrant's contact information be visible in WHOIS. Where the registry permits private registrations, though, the proxy contact information will continue to be associated with our Brea postbox, just as existing ones are. Nothing changes in this regard.
Domain registrations which require non-private registration, or which require additional information for registration (e.g, .ASIA) will specifically ask you for this when you attempt to register the domain, as we're aware that this will not necessarily be acceptable for all users.
If you want to see an example, you can check out the WHOIS information for "securityfla.ws" (which I have registered with private WHOIS). The exact details how the WHOIS information appears will differ from one registry to another, as every registry runs their own separate WHOIS servers.
.US is probably a bad example here — I don't recall offhand which registries have this requirement, but I seem to remember a few of the European registries being among them.
For what it's worth, the address of the registrar (e.g, New Dream Network or Directi) will generally not show up in domain registrations. None of the WHOIS entries (for any TLD) I checked had anything more extensive than name/email/phone, and .US specifically looks like this:
Sponsoring Registrar: DIRECTI INTERNET SOLUTIONS PVT. LTD. D/B/A PUBLICDOMAINREGISTRY.COM
Registrar URL (registration services): www.publicdomainregistry.com