We can first pick some low hanging fruits based on clever observations.
Looking around, the devoicing marker ◌̥ appears on and only on the final vowels, with the exception of [i] in "dii"—this is also consistent with all Ik sentences we are to see. Therefore, this marker is purely phonological and we can safely ignore it for analysis. It's also the answer for E6.
C. "We came from Ethiopia." and H. "Father is coming from Ethiopia." should match to something with "Ethiopia"; looking around, we see 9. "Atsima bee Isopiao̥." and 16. "Atsa abaŋa Isopiao̥.". It even nicely capitalizes "I" for us. However we don't know which is which yet.
I. these huts and O. the huts are non-sentences and must match to 14. "hoikḁ" and 19. "hoika dii". The demonstrative form is nearly always more complex, so 14 = O, 19 = I, and "dii" = "these", "hoika" = "huts".
11. "hoika dii ɲcie leɓetse ni gaanḁ" must match to B. "these two huts of mine that are bad" which is the remaining non-sentence phrase and has "these huts".
2. "Atse̥." and F. "Come." are both single-word sentences, so they must match (there's no other sentence that expresses a single atomic idea). So "atse" = "come (imperative)".
We have 4 other sentences containing "ats-": 9 and 16 as identified above, 5. "Ats’a ŋoka ɔkakḁ.", and 7. "Atsia hoo̥.". 9 and 16 are already known to match to something else, so one of 5 and 7 must match the remaining sentence with "come": Q. "I am coming from the hut." Since "huts" is "hoika", "hoo" looks more likely to be "hut". So 7 = Q.
"hoo" appears in 15. "Epa bee ŋoka hoo̥.". The other "hut" sentence is E. "The dog slept in the hut."
A. My dog is sleeping in the shade. D. The dog is sleeping in the nice shade. G. He loves coffee. J. The dog that is white is sleeping. K. I love tea. L. Go sit. M. The elephant is sleeping. N. He loves his wife. P. Father went. R. The dog is chewing the bone. S. He is going with the elephant.
9. Atsima bee Isopiao̥. 16. Atsa abaŋa Isopiao̥.
C. We came from Ethiopia. H. Father is coming from Ethiopia.
The word "ŋoka" repeated 4 times, in 3, 5, 8, 13. The word "epa" also repeated 4 times, in 3, 8, 13, 17. The idea that repeats 4 times are "dog" and "sleeping", in A, D, J, R, and A, D, J, M respectively. So A, D, J, map to 3, 8, 13, while R and M map to 5 and 17, in some order. Because M. "The elephant is sleeping." is two ideas but R. "The dog is chewing the bone." is three ideas, we can guess that 5 = R, 17 = M based on their word count. Therefore, "ŋoka" = "dog", "epa" = "sleeping", "elephant" = "oŋora".
"oŋoro" appears in 12. "Ƙaa oŋoro̥.", so this is S. "He is going with the elephant.", so "Ƙaa" = "he is going". The other two sentences starting with "Ƙaa" (you should probably have noticed that the verb is at the beginning at this point), 1. "Ƙaa bee abaŋḁ." and 6. "Ƙae zeƙwe̥.", must be L. "Go sit." and P. "Father went.", in some order. Recall from the very beginning, we have one of 9. "Atsima bee Isopiao̥." and 16. "Atsa abaŋa Isopiao̥." map to H. "Father is coming from Ethiopia." We have two shared words among 1, 6, 9, 16: "bee" and "abaŋa". However, "bee" appears in 15. "Epa bee ŋoka hoo̥." = E. "The dog slept in the hut.", so it cannot mean "father" (and is more likely the past tense morpheme). Therefore, "abaŋa" = "father", and 1 = P, 6 = L, 9 = C, 16 = H.
Ik
English
4. Minia ɲecayḁ. 10. Mina cekia ntsie̥. 18. Mina ɲakawakḁ.
G. He loves coffee. K. I love tea. N. He loves his wife.
3. Epa ŋoka na ɓets’ḁ. 8. Epa ŋoka ɲcie kuruo̥. 13. Epa ŋoka kuruo na daḁ.
A. My dog is sleeping in the shade. D. The dog is sleeping in the nice shade. J. The dog that is white is sleeping.
The remaining ones need some more careful analysis.
In the "love" group, we have two "he loves" and one "I love", and two "Mina" and one "Minia". So 4 = K, and "ɲecaya" = "tea" (reasonable since it's derived from "chai"). The other two-word sentence, 18, is G, with "ɲakawaka" = "coffee" (reasonable by pronunciation). That leaves 10 = N.
In the "sleep" group, we have two sentences containing "kuruo" and two sentences containing "shade", so "kuruo" = "shade", and the remaining one 3 = J. The word "ɲcie" in 8 can either mean "nice" or "my". It also appears in 11 = B. "these two huts of mine that are bad", so it has to mean "my". Therefore, 8 = A, 13 = D.
Having studied all the word correspondences, we can now say a few things about word order: this language is VSO, with adjectives, demonstratives, and relative clauses coming after the noun. The verb agrees with the subject in person and number. If the subject is a pronoun, it can be dropped. The past tense morpheme ("bee") comes after the verb.
1. Epa ŋoka kuruo na daḁ. = sleeps dog in-shade which is-nice ("na" appears in 3 "na ɓets’ḁ" = "that is white" and 13 "na daa" = "nice")
2. Epa ŋoka na ɓets’ḁ. = sleeps dog which is-white
a. these = dii
b. my huts = huts my = hoika ɲcie̥ (the other one is "hoika na ɲcie̥", but I don't know how you are supposed to see that "na" also works with possessives because the cases we've seen of "na" are all with adjectives.)
c. I love my wife. = love-1sg wife my = Minia cekia ɲcie̥. (again, the other one is " Minia cekia na ɲcie̥.")
d. Father is coming from the nice hut. = come-3sg-prog father hut which is-nice = Atsa abaŋa hoo na daḁ.
a. Zeƙwata oŋorika kuruo̥. = sit-3pl elephants shade = The elephants are sitting in the shade. (We haven't seen the first two words exactly, but we've seen "zeƙwe" and "oŋoro" with the same roots. Based on "hoo"-"hoika", we can infer that "-ika" is the plural suffix; there aren't many other things you can attach to a noun anyway.)
b. Mina ŋoka ɔkaka ntsie̥. = love-3sg dog bone his = The dog loves his bone.
c. Minima oŋorika ni epḁ. = love-1pl elephants that sleep = We love elephants that sleep. (The solution also gives "We love sleeping elephants." but this is really hard to think of because they are meaning-wise equivalent. The "-ima" suffix means 1pl based on 9 "Atsima".)
d. ŋoka na ɲcie̥ = dog which my = my dog (Why doesn't the answer say "the dog of mine" as well? I'm not sure.)
In E5:
hoika dii ɲcie leɓetse̥: we've seen this sequence verbatim in 11, so this is grammatical.
hoika dii ɲcie ni leɓetse̥: also grammatical, although we don't have very solid proof. This would be "these huts of mine which are two". It seems that all noun modifiers are equivalent and can either follow "ni" or not.
Based on this observation, in E7, our answer is that adjectives are not a separate class because adjectives are embedded in relative clauses (in which case it's equivalent to a verb). Any sentence containing a relative clause with a single adjective is plausible as answer: 3 = J "The dog that is white is sleeping.", 11 = B "these two huts of mine that are bad", 13 = J "The dog that is white is sleeping.".