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This answer in its present form states

If Donald Trump were impeached and blocked from running again (which is one of the things that impeachment can do, disqualify from office)...

Is that true?

Does impeachment by the House of Representatives, with or without subsequent removal from office by the Senate, actually disqualify one from being re-elected to the same Presidential office?

Before the present time, this might have seemed politically impossible regardless of the law, but impeachment and even removal from office does not seem as likely to hinder the viability of a candidate supported by the base of voters supporting the current US President. This question is intentionally on Law.SE instead of Politics.SE as the focus is intended to be on law rather than political viability.

As an example of a similar case, consider Roy Moore, who in 2001 was elected as Alabama's 27th Chief Justice, removed from office in 2003, was again elected chief justice in 2013, again forced out, then won a Republican primary for Senate in a heavily Republican state, earning Trump's endorsement, and is again a candidate for the US Senate.

As a closely related question, does impeachment by the House of Representatives, with or without subsequent removal from office by the Senate, actually disqualify one from being re-appointed as Vice President or elsewhere early in the line of succession, and re-reaching the same Presidential office by resignations or otherwise?

Impeachment of a president does not on conviction automatically disqualify the convicted party from becoming president again. However, after conviction, the Senate can vote to add to the punishment of removal from office "disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States." This requires only a simple majority vote.

Does impeachment by the House of Representatives, with or without subsequent removal from office by the Senate, actually disqualify one from being re-elected to the same Presidential office?

See above. It requires impeachment by the House, conviction by the Senate, and a separate vote by the Senate to impose the punishment of disqualification.

It's conceivable that the Senate could disqualify the convicted party only from becoming president, though it looks like in the two prior instances where this punishment was imposed it was the broader disqualification. But if a person disqualified only from being president is in an office that would normally be in the line of succession, that person is simply omitted from the line of succession. This happens routinely with naturalized citizens, and there's no reason to think it would be any different for a former president who had been disqualified only from the office of the president after being convicted on articles of impeachment.

Impeachment by the House does not legally disqualify someone from office, only the Senate can vote to "disqualify" an officeholder from holding any public office again, and only after a successful conviction/removal.

Here's the relevant passage from Article 1, Section 3, Clause 7 of the Constitution (emphasis mine):

Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.

This has been interpreted to mean that disqualification is a separate vote, and that it is only possible after a successful vote to remove. From Heritage.org:

Since ratification, four troublesome questions have arisen under this clause. The first was whether the Senate may impose the sanctions of removal and disqualification separately and, if so, how. The Senate claims that it may impose these sanctions by separate votes: (1) removal, involving the ouster of an official from the office he occupies at the time of his impeachment trial, and (2) disqualification barring the person from ever serving again in the federal government. In 1862 and 1913, the Senate took separate votes to remove and disqualify judges West Humphreys and Robert Archbald, respectively. For each judge, a supermajority first voted to convict followed by a simple majority vote to disqualify. The Senate defended this practice on the ground that the clause mentioning disqualification does not specify the requisite vote for its imposition, although Article II, Section 4, mentions removal as following conviction. The Senate in 1862 and 1913 considered that the supermajority requirement was designed as a safeguard against removal that, once satisfied, did not extend to the separate imposition of disqualification.

This is incorporated in the current US Senate overview of the impeachment process (PDF):

The Senate may subsequently vote on whether the impeached official shall be disqualified from again holding an office of public trust under the United States. If this option is pursued, a simple majority vote is required.

And in the official PROCEDURE AND GUIDELINES FOR IMPEACHMENT TRIALS IN THE UNITED STATES SENATE:

Following the Vote on Each Article, the Presiding Officer Pronounces the Decision. Once the Judgment of the Senate has Been Pronounced on the Articles of Impeachment, the Trial Might Progress in Two Ways. If the Respondent Was Found Not Guilty on All Charges, the Verdict of Acquittal Was Announced and the Senate Sitting as a Court of Impeachment Adjourned Sine Die. If the Respondent Was Found Guilty of Any of the Charges, the Judgment of Removal and Possible Disqualification From Ever Holding an Office of Trust or Profit Under the United States Was Presented.

This lengthy document also contains details about the few times this power of "disqualification" was used.

Regarding the remainder of your question:

I don't see any provision in any of these sources that the Senate could selectively disqualify someone from some offices and not others. They all use the phrase directly from the constitution, "disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States".

However, the phrase "Office of honor, Trust or Profit under the United States" may not be quite as obvious as it appears. This 2014 law review article claims the disqualification clause may not apply to elected positions, especially to Congress: 'You've Got Your Crook, I've Got Mine': Why the Disqualification Clause Doesn't (Always) Disqualify

I don't have the ability to weigh in on the seriousness of this claim, but there is extensive (and ongoing) discussion on this Congressional legal blog (see further related discussion under the disqualification and the office tags).

There is at least one currently seated House Rep who was previously a Federal Judge who was impeached by Congress but not disqualified from future office in US government. In the case of Moore, a state legislature's impeachment of an office holder may bar him from future state office but no longer has the power to block the person from federal office (originally, U.S. Senators were indirectly elected by the legislature of their state, and would represent state government interests. This was changed by the 17th Amendment.).