In a discussion thread elsewhere, someone mentioned that Stephens died as governor of Georgia. I was curious about that because I knew that Section 3 of the 14th Amendment says that anyone who took an oath to support the Constitution of the U.S. and then participated in a rebellion against the U.S. is banned from holding office. This would apply to Stephens, unless I misunderstand something.
Stephens' Wikipedia article mentions that he was elected to the Senate in 1866 but forbidden to take his seat "because of restrictions on former Confederates." (This is odd because it's before the adoption of the 14th Amendment. What restrictions were in place in 1866?) But then it says he was a member of the House of Representatives from 1873 to 1882 without explanation of how he got around the 14th Amendment.
The 14th Amendment allows former rebels to hold office if Congress votes by a 2/3 majority to allow them to do so. Did this happen with Stephens? Or did Congress fail to enforce the Amendment? When Stephens was first elected to the House, Reconstruction was still ongoing, so either alternative seems worthy of further explanation.
Did other former Confederates hold elected office?
The simplest reason he could enter Congress is the Amnesty Act of May 1872, and President Grant's proclamation of 1 June 1872 that suspended the prohibitions on those grounds. Stephens had received a pardon in 1865, and general amnesty later in a few waves created conditions where that could happen. Stephens and his correspondents talk about this in some of his papers but I can't find the precise ones because the Stephens digital collections at the LoC and Duke are only indexed by box, not sender, topic, or even exact date, though they are at least in order. I think there are other sub-collections out there, with more of his own writing, but I've not seen them yet. The Amnesty Papers lay out Stephens's case for a pardon but I don't have a link to those; J. T. Dorris's ancient 1928 article "Pardoning the Leaders of the Confederacy" gets into the variations and pardons/amnesty under Johnson, though there's surely newer and better critical discussion out there. (Sorry, paywall.) Of course, a pardon or amnesty alone wouldn't remove the 14th amendment restriction, so the Act of 1872 (which was recorded as passing with the requisite 2/3 vote after brief discussion/) was still necessary to achieve that goal.
Whether or not other Confederates served, they could, after mid-1872. The big issue over Stephens's first (1866) rejection arguably had to do more with Georgia not yet having been properly or fully re-admitted to the Union, and not so much the man himself (!!). It's hard to say whether he'd have survived a Congressional vote without that procedural restriction foreclosing it, but it made everyone more comfortable not to be confronted with the specter of such a personal vote either way.
[edits: added links to the Stephens digital papers, cited Dorris because it's one I've seen before when working on the CSA Dept of State; found a source for the 2/3 vote being met in Rawley's 1960 note linked above. I really haven't just wandered into the wrong flair neighborhood, honest!]
In terms of the last question, did other former Confederates serve, the answer is yes. I'll give just a couple of examples. Allen T. Caperton, a CSA Senator for Virginia, in a remarkably interesting twist served West Virginia in the US Senate from May 1875 until his death the following year. Confederate Gen. William Cox, more notably, became Secretary of the Senate in 1893, in symbolism the Senate itself notes.
I've never seen a full run-down, and it's clear that most former representatives and officers of the Confederacy did not pursue an active public (read: political) life after the war. But the option was there.