If John is a dinosaur, he will say that a dinosaur is a dinosaur (true) and an elephant is a dinosaur (lie.) If John is an elephant, he will say that an elephant is an elephant (true) and a dinosaur is an elephant (lie.) Therefore all we can establish from John says "Zayin is a dinosaur." is that John is a dinosaur.
There are 16 combinations of animals for the second part. For example TD JD MD ED would mean all 4 are dinosaurs. Since they would each tell the truth, starting with M saying "E is a D", you end up with T saying "E is a D" which is not what was said, so that combination can be ruled out.
There are more than 1 combinations that lead to the sentence being uttered though: for example TE JD MD ED would make M say "E is a D" truthfully, and J pass that along truthfully, but then T lie and report the chain ended E. Similarly TD JE MD ED would start with M saying "E is a D" but then J would lie, claiming M said E was an E, and then T would pass that along truthfully. I haven't tested all 16 combinations, but my suspicion is that whenever you have an odd split (1-3 or 3-1) of animal types, that is the sentence that will be uttered.
For the third part, the sentence "you lie about your righthand neighbour" involves 3 animals: the speaker S, the subject Y, and the neighbour N. There are 8 combinations possible: DDD, DEE, DED, DDE, EEE, EED, EDE, EED. In the 4 cases where Y and N are different, Y really does lie, but only if S and Y are the same will S say so. In the other 4 cases, Y tells the truth, but if S and Y are different S will lie about that. So the patterns DEE, DDE, EED, and EDD will lead to this sentence. You can get this pattern if they sit in pairs - for example four of them as DDEE with the final E "looping back around" and sitting next to the first D. The total number of animals must be a multiple of 4 to reuse this pattern, eg DDEEDDEE and so on.