Question

Is 10,11,10,00 a legit gray code? How about one that is not cyclic (first term 00000, last term 11111)?

Answer 1

10, 11, 10, 00 yes.
I dont understand your second question.
Bibliography: http://pages.cs.brandeis.edu/~storer/JimPuzzles/MANIP/ChineseRings/READING/GrayCodesWikipedia.pdf

Answer 2

But, does this condition on Wikipedia not apply?
"Gn is a permutation of the numbers 0, ... , 2n−1. (Each number appears exactly once in the list.)"

Answer 3

Gray code has to be cyclic (difference only at one position between any two numbers), 00000 and 11111 has 5 differences. It's useful for http://en.wikipedia.org/wiki/Karnaugh_map