How did Nakamoto untraceably pay for registering bitcoin.org? Wikipedia says &quot;Nakamoto created a website with the domain name bitcoin.org&quot; and &quot;transferred several related domains&quot;, while bitcoin.org says it was &quot;originally registered and owned by Bitcoin's first two developers, Satoshi Nakamoto and Martti Malmi&quot;.  A 2014 businessinsider.com article says bitcoin.org was first registered in 2008 via anonymousspeech.com, and that Malmi stated bitcoin.org was &quot;originally created by Satoshi&quot;, and Malmi's first contact with Nakamoto was in 2011 when bitcoin.org was transferred to Malmi.
The article also says anonymousspeech.com accepted cash and E-gold.  Did Nakamoto pay for the original bitcoin.org registration, and used cash or E-gold (paid for with cash) to do that and keep his origins untraceable?  Or is it possible that somewhere in history is a non-anonymous payment of some kind that would finger Nakamoto?  For example, archive.org shows bitcoin.org was an actual hosted site at least by January 31, 2009.  Where was it hosted, and how and to whom was that hosting paid for?
How did Nakamoto untraceably pay for registering bitcoin.org?

Nobody knows for certain apart from Nakamoto and the businesses involved.
Nakamoto was (and remains) notably secretive and their true identity is not reliably known† and they no longer communicate with the world using that name in any verifiable manner.
All the businesses involved in the alleged registration process protect the identity and other confidential information associated with their customers. As they should. Therefore the details sought are not public knowledge.


first registered in 2008 ...

That certainly appears to be the case.
The domain registry information currently includes the following information:
Domain Name: BITCOIN.ORG
Registry Domain ID: D153621148-LROR
Registrar WHOIS Server: whois.namecheap.com
Registrar URL: http://www.namecheap.com
Creation Date: 2008-08-18T13:19:55Z
Registrar: NameCheap, Inc.


...via anonymousspeech.com,

The website of AnonymousSpeech.org says

Any inquiries  regarding the identity of our subscribers are ignored. We do not respond to any of them.


We are committed to the privacy of our users, and will absolutely not release any kind of data to any foreign governments.



archive.org shows bitcoin.org was an actual hosted site at least by January 31, 2009. Where was it hosted, and how and to whom was that hosting paid for?

That information is not available.
We know a couple of important things about Nakamoto:

They were extremely protective of the secrecy of their identity.
Though their work was not without flaws, they were competent in the field of online privacy and encryption.

We should therefore not expect to easily find any information they specifically sought to hide from the public.
Nakamoto's main aim was to create a cash system that did not rely on trusting third parties - their aim was not to teach us how to anonymously register DNS domains. This should shape our expectations of available information.

Footnotes
† So far as I know, the people claiming to be Nakamoto have failed to provide proof or have provided evidence subsequently shown to be false. We cannot ask Nakamoto and if we could, they would not an answer if their behaviour is consistent with their past behaviour.