Ok now I really should sleep, though working on the rest is tempting. I need to rest my shoulder from all the typing.
Fast method for anyone that wants a head start tomorrow, take the "accounts for a change of x from the shuffled position." and add that to the "y additional" for each of its options. If x + y does not equal at LEAST five, then that entire subgroup is impossible. The last 2 x's in the position can only have a change of 10 and the total change needs to be 15.
Example:
10. xxx89 change of 2
10.1 one additional (2 + 1 = 3) IMPOSSIBLE
10.2 zero additional (2 + 0 = 2) IMPOSSIBLE
10.3 four additional (2 + 4 = 6) POSSIBLE
10.4 two additional (2 + 2 = 4) IMPOSSIBLE
10.5 four additional (2 + 4 = 6) POSSIBLE
10.6 three additional (2 + 3 = 5) POSSIBLE
Sorry, forgot to exclude the rule that we know none of the 01234 was in the correct string. Please do consider forgiving me for my error.
1.
56822
56495
56592
56693
56527
2.
53793
55726
3.
53683
55683
54486
4.
55629
5.
86722
16751
26755
56725
26728
6.
96383
36783
26782
16585
26685
36385
36585
46485
7.
26599
26559
96529
56529
26829
8.
25785
9.
15719
10.
25589
32 possibilities left.
Here are the basic principles of what I guess.
1. First Guess every number +3 and then every number -3, this is a possible combination and feels likely at least one will pop up. If one of these results gives over 4 numbers, then you just move them around while making valid combinations and slotting in a fifth number.
2. Next think go for a group you think has the remaining numbers, if very few of the +3/-3 appear, then you know that there need to be a good number of +/- 4s or 5s, since the only combos that add up to 5 are 4/1,3/2 and 5/0 and the lack of 3s means the only high numbers can be 4s and 5s. You should only need to guess one of these, maybe 2. Always go for the one with the most different digits.
3. From these the number correct will tell you the digits, and the in the right place will tell you the rough composition of the sum so far. Then you just start shuffling around with a guess on what you think happened but be careful to not have contradictory ideas (i.e if you know 3 of the first guess are in there, don't guess 4 of those numbers).
It's not super well defined but that's the jist.