EDIT: not related to reconnect link (the reconnect bug may cause adouble review, which will jump a card two stacks.

If you're seeing cards twice, it is most likely that your session expired, or you closed the browser/window without going to the review summary. You must click "End" to make sure all card answers are "flushed" to the server. See here

If a card is in the final box, and is reviewed correctly, shouldn't it be moved to the end of the final box, more or less?

No, there is a randomize interval, which is wider for the last boxes than the first. That's because you can't randomize 3 day base interval on 30 days, and it won't be very helpful to randomize the 6 month delay over a few days.

Still, base interval + random interval (to spread cards) should never reschedule a card on the same day, not even remotely close to the scheduled date of the last review; because each interval is at least twice that of the previous one, and the spread intervals (random amount) maximum negative (and positive) adjustment is between 1/3 (lower box) to 1/8 (highest box) of the base interval.

The higher base interval (not counting +50% "Easy") is 240 days, with a spread of -30 to +30 days. For 3 days base interval (second box), the spread is only -1 to +1 days, so the minimum interval is 2 days. So even if you reviewed before and after your timezone setting-based "midnight" time, you can't see the same card during reviews.

It shouldn't happen that you review twice the same flashcard, on the same day (understanding "day" where the zero hour is based on your Profile's time zone setting); because minimal interval is two days (including reviewing "relearned" pile from the Study page).

For what it's worth, I went through and carefully recorded all the card numbers.  Here is the result of today's study:

Initial state of my cards:
Box 4: 964, 1800, 2030, 2188, 2562
Box 6: 1444, 1634, 2051, 2207
[everything else in Box 8]

First review of the day, I saw these cards in order("*" = missed):

1800, 2030, 2562, 964, 2188 (all from Box 4),
2051, 1444, 2207, 1634 (all from Box 6),
And then I got, from Box 8 (divided in groups of 10 for later reference):
(A) 55, 691, 450, 276, 1748, 610, 2235, 207, 2905, 2933,
(B) 46, 792, 320, 2281, 1537, 1194, 403, 2359, 2450, 2833,
(C) 2808, 852, 2719, 1309, 1166, 2534, 1833, 1202, 1519, 2887,
(D) 160, 1798*, 2322, 2097, 1453, 1812, 1737, 2728*, 1928, 502,
(E) 1819, 112, 1621, 183, 2164, 374, 1881, 1963, 2702, 444,
(F) 1579, 1509, 613, 2701, 1451, 2578, 2332, 181, 2911, 2867,
(G) 2696, 2380, 1921, 2137, 2901, 108, 1754, 1153, 1373, 1684*,
(H) 841, 457, 1741, 2370, 890, 901, 1562, 571, 2490, 935,
(I) 1552, 2714, 1923, 779, 2935, 437, 299, 2402, 2962, 847,
(J) 2026, 2227*, 2058, 2715, 1273, 2309, 185, 677, 480, 1859,
(K) 2566, 201, 516, 654, 363, 864, 376*, 372, 2772, 2091,
(L) 581, 2301, 2975, 2859*

I stopped there because I'd exceeded 100 reviews and missed my sixth one for the day.  I carefully clicked on "End", went back and reviewed/cleared the failures.  Then I returned to the "review fresh" link, and here are the cards that I was given, in the order I saw them:

376, 2728, 1798, 2227, 2859, 1684 (the six I missed today),
2562, 1800, 2030, 2188, 964 (the five that started the day in Box 4),
2207, 1444, 2051, 1634 (the four that started the day in Box 6),

So far, that's what is expected.  The just-failed cards should be scheduled for review almost right away.  The ones that started in Box 4 and were just moved to Box 5, should be due next.  The ones that started in Box 6 and were just moved to Box 7 should be before any of the ones in Box 8 (since I turn over Box 8 so quickly).

And then I got, from Box 8:

1833 (C-7 above), 444 (E-10 above), 2450 (B-9 above), 2322 (D-3 above),
792 (B-2 above), 2719 (C-3 above), 2605, 809, 737, 451, 2529, 1018,
1938, 1207, 2214, 2455, 1155, 896, 2128, 527, 865, 884, 950, 1538,
2512, 2427, 1608, 1219, 1604, 904, 2462, 557, 1624, 2697, 2644, 2585,
1945, 2139, 422, 1191, 369, 760, 1954, 2916, 2917, 1347, 2932, 2850,
2820, 956, 554, 390, 2198, 1536, 2179, 2382, 2103, 122, 2010, 2652,
1445, 2204, 669, 952, 415, 794, 2804, 2976, 293, 1526, 1237, 2307,
1009, 2316, 493, 297, 1854, 1828, 153, 1905, 1502, 955, 1716, 324,
1275, 2247, 2436, 123, 1509 (F-2 above).

When I got up to Box 8, the first six cards I saw where ones that I had just reviewed from Box 8.  Then I had to go through almost 100 more cards before I saw the seventh repeat from the earlier group of Box-8 reviews.

The six up-front suggests to me that it is an artifact of the rate at which I cycle through Box 8.  So... I don't think there's any general issue with the sort ordering, it's my fault for not using the card boxes "properly" (waiting for cards to expire).