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).