Is a separate law needed to prohibit people to whom Section 3 applies from taking office, or to remove such people from office?
There is some reason to think so. In 1869 there was Griffin’s Case, 11 F. Cas. 7, 2 Am. Law T. Rep. U. S. Cts. 93; 8 Am. Law Reg. N. S. 358. This was tried by Chief Justice Chase, in his role as a Circuit Justice, not as a Supreme Court case.
One Caesar Griffin was accused of shooting with intent to kill, tried before a Virginia state court, convicted, and sentenced to two years in prison. He then filed a writ of Habes Corpus claiming that his imprisonment was unlawful, because the jusge before whom he was tried, Hugh W. Sheffey, (in the words of Justice Chase):
in December, 1849, [Sheffey] as a member of the Virginia house of delegates, took an oath to support the constitution of the United States, and also that he was a member of the legislature of Virginia in 1862. during the late Rebellion, and as such voted for measures to sustain the so-called Confederate States in their war against the United States; and it was claimed in behalf of the petitioner, that he thereby became, and was at the time of the trial of the petitioner, disqualified to hold any office, civil or military, under the United States, or under any state ...
Several other people tried and convicted before Judge Sheffey in the circuit court of Rockbridge county, including some convicted of murder, filed similar Habes Corpus petitions, claiming that Sheffey was prohibited by Section 3 from holding office, and that therefore their convictions were invalid. Judge Sheffey had been appointed to the office of Judge after the end of the US Civil War by the reorganized government of Virginia, the one recognized as valid by the Federal Government. He was appointed before the ratification of the 14th Amendment. There was no dispute that he fit the letter of the group of prohibited persons in Section 3.
Justice Chase wrote:
The general question to be determined on the appeal from this order is whether or not the sentence of the circuit court of Rock-bridge county must be regarded as a nullity because of the disability to hold any office under the state of Virginia, imposed by the fourteenth amendment, on the person, who, in fact, presided as judge in that court.
Justice Chase rejected the suggestion that all official actions by Sheffey and anyone in a similar position were automatically void and of no authority because of sectio0n 3. He wrote that:
The proposition maintained in behalf of the petitioner, is, that this prohibition, instantly, on the day of its promulgation, vacated all offices held by persons within the category of prohibition, and made all official acts, performed by them, since that day, null and void.
After pointing out that many of the offials of the reconstructed governments of the sothern states were withign the terms of the prohibition of section 3, Chae went on to write:
If the construction now contended for be given to the prohibitive section, the effect must be to annul all official acts performed by these officers. No sentence, no judgment, no decree, no acknowledgment of a deed, no record of a deed, no sheriff’s or commissioner’s sale — in short no official act — is of the least validity. It is impossible to méasure the evils which such a construction would add to the calamities which have already fallen upon the people of these states.
As a further reason not to construe section 3 as instantly and automatically removing all such officials, Chase wrote:
Now it is undoubted that those provisions of the constitution which deny to the legislature power to deprive any person of life, liberty, or property, without due process of law, or to pass a bill of attainder or an ex post facto, are inconsistent in their spirit and general purpose with a provision which, at once without trial, deprives a whole class of persons of offices held by them, for cause, however grave. It is true that no limit can be imposed on the people when exercising their sovereign power in amending their own constitution of government. But it is a necessary presumption that the people in the exercise of that power, seek to confirm and improve, rather than to weaken and impair the general spirit of the constitution.
Is there, then, any other reasonable construction? ... The object of the amendment is to exclude from certain offices a certain class of persons. Now, it is obviously impossible to do this by a simple declaration, whether in the constitution or in an act of congress, that all persons included within a particular description shall not hold office. For, in the very nature of things, it must be ascertained what particular individuals are embraced by the definition, before any sentence of exclusion can be made to operate. To accomplish this ascertainment and ensure effective results, proceedings, evidence, decisions, and enforcements of decisions, more or less formal, are indispensable; and these can only be provided for by congress.
[I]t seems to put beyond reasonable question the conclusion that the intention of the people of the United States, in adopting the fourteenth amendment, was to create a disability, to be removed in proper cases by a two-thirds vote, and to be made operative in other cases by the legislation of congress in its ordinary course.
Chase also note that, 2 months after the conviction of Griffin:
in February, 1869, congress adopted a joint resolution entitled “a resolution respecting the provisional governments of Virginia and Texas.” In this resolution it was provided that persons, “holding office in the provisional governments of Virginia and Texas,” but unable to take and subscribe the test oath prescribed by the act of July 2, 1862 [12 Stat. 502], except those relieved from disability, “be removed therefrom;”
and that this indicates that such people were regarded by Congress as having remained in office and not beign automatically removed.
Griffin's Case seems to establish that some proceeding is needed to establish when a person falls under the exclusion rule of Section 3 of the 14th. In 1869 this was done by military order. In 1870 Congress passed a statute providing for criminal proceedings in such cases. This was repealed at the end of Reconstruction.
Currently 18 U.S. Code § 2383 makes "rebellion or insurrection against the authority of the United States or the laws thereof," a crime, and provides that anyone convicted of it "shall be incapable of holding any office under the United States." This is not quite the same as the section 3 disqualification. One the one hand, it does not depend on a prior oath to support the constitution, and on the other it does not appear to ban holding a state or local office. But it shows how a similar law could be drafted by Congress.
Can a Congressional resolution, or a Federal statute declare that specific people have so engaged?
This is probably prohibited as a Bill of Attainder, that is, a legislative declaration, without trial, that particular persons are guilty of particular crimes.
What proceedings would be needed to establish that a person had "engaged in insurrection or rebellion"?
If a person subject to the section 3 disqualification was elected to either house of Congress, that house could refuse to seat such person, or expel him or her, as the Constitution permits. Most state legislatures have similar powers over the seating and expulsion of their own members.
If a person is convicted under 18 USC § 2383 or any similar law, that person would clearly be excluded. Congress could pass a law establishing a special tribunal for determining when a person was subject to Section 3 of the 14th. But it has not done so.