Thought I'd weigh in a bit on the Tillman puzzles, which Mike Ramsey discussed here. On the civil/military and "office under the United States" issues, 14A/3 seems relevant. The oath-denying-Confederate-disqualification provision begins, "No person shall be a Senator or Representative in Congress, or elector of President and Vice-President, or hold any office, civil or military, under the United States, or under any State…" Unless "civil or military" is redundant, "civil" does seem to be "non-military," though like Mike I find the no-impeachment-of-military-officers and Congressmen-allowed-to-have-military-offices-simultaneously consequences under II/4 and I/6/2 to be odd.
Also, if Confederate disqualification extended to the Presidency and Vice-Presidency, which it seems quite plausible that it did, they'd have to be "office[s] … under the United States," I think, pushing me towards Mike's view on puzzle 5.
[Update: Seth writes to mention Sai Prakash's use of the same 14A/3 argument here at 151. Seth notes, however, that section 4 was not particularly well-drafted; John Randolph Tucker argued that "office … under any State" seems to exclude members of state legislatures, though they are part of the trigger for the disqualifiation. See here at 69-71 n.119. And of course 14A/3 and I/6/2 might use the term differently.]
Another reason pushing me in that direction in the interpretation of I/6/2 is the combination of II/1/7 and I/6/1: the President is not to receive any additional emoluments during his time in office, but representatives and senators are to receive a salary. Of course, that might simply be an independent reason why the President can't be a representative or senator, rather than a reason to read I/6/2 to ban that result. It also wouldn't, on its own, apply to the Vice President.
[Update: Seth writes to remind me that I made the same point in an email exchange with him in 2010. His response then: "[T]he specific language in regard to the President (Article II) trumps the more general language in regard to members (in Article I). Also, a tension in regard to emoluments even if problematic does not work an incompatibility. Incompatibilities, historically, must be express; they do not arise by implication."]
[Update 3/13: Just realized I'd cited 14A/4, the debt provision, rather than 14A/3, the oath-denying-Confederate-disqualification provision, so I changed it above.]
Posted at 5:23 PM