A couple of questions.

[True Federalist (진정한 연방 주의자)]

2) Can a president who never served as vice-president run for vice-president after their terms as president have expired?

From the XIIth Amendment:
"But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."

From the XXIInd Amendment:
"No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once"

You'd have to cook up some very convoluted logic to argue that a person who is ineligible to be elected to the office is somehow still eligible to the office.  It certainly woud go against the clear intent of the XXIInd.

However, I think it would be possible for such a person to become acting President under the XXth Amendment in the case of a failue of either a President or Vice President to be elected, who would serve until one or the other office were to be filled.

[True Federalist (진정한 연방 주의자)]

I should point out, however, the Constitution is completely silent on succession beyond the VP position, which might be the reason it isn't clearer.

Not completely silent, merely mostly. Atricle II Section 1 Clause 5 states:  "In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected."

The XXth and XXVth set the procedure for when the Vice President takes over and what happens when there is a failure to be either a P-elect or a VO -elect, but the Constitutional grant of authority of what to do when there is no President or VP goes back to the beginning

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Even if one holds that it's not in the constitution, it is spelled out in 3 USC 19 that the Speaker or President pro tem would have to resign in order to become either President or acting President.

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Actually, W's already ineligible and has been since January 20, 2003 to be elected again.  The prohibition against him or Clinton becoming acting President is legislative rather than Constitutional, since Congress does have authority to allow persons elected twice to serve as acting President, it merely has chosen to not so allow.  For example, I could see the law be changed to allow a former President to serve as acting President for a day if we got another President who refused to be sworn in on a Sunday.

[True Federalist (진정한 연방 주의자)]

Might I ask, what is the basis for this assumption? What is the basis for the claim that, when the Twenty-Second Amendment was written, eligibility for the Presidency equaled eligibility to be elected President, when the amendment itself differentiates between being elected and succeeding?

It deals with the qualifications for holding the office.  The XII Amendment makes the requirement that the VP be constitutionally eligible to be President, when chosen.  GWB or WJC could not be eligible, if chosen for VP.  This would be like saying that a 25 year old could be slipped in as president

Prior to the XIIth, the Vice President was the person who came in second in the Presidential election, and as such had to be inherently eligible to be elected President.  With a separate ballot for the Vice President, the qualifications had to be explictly stated for that office as well.