Q1.13(a) You are justified to be concerned. As stated in the book, we have a problem with reflexivity (a bundle is at least as preferred as itself) and the indifference set being empty. This was pointed out in an earlier posting by Bettina Klaus. The usual definition of a lexicographic ordering is in strict preference if strict inequalities are used. With this definition the following two quotes may be helpful. 'The bundles are ordered according to the principle of words in a dictionary or lexicography...In this case, each bundle has no points (other than itself) to which it is indifferent; indifference surfaces cannot be drawn and no utility function exists'. (Deaton and Muellbauer, Economics and consumer behavior (1980)). Or look up the entry by C Blackorby on lexicographic orderings in The New Palgrave: 'There are no trade-offs between [bundles] a and b and each indifference set is a single point'.
So the answer is a single point using the usual definition of a lexicographic ordering.
(b) For a proof see Debreu (1954), footnote 1 as referenced on page 531 of the book. The paper is also in Debreu's book 'Mathematical Economics'. Although the footnote is short, it is not easy! The proof in Mas-Colell, Whinston and Green 'Microeconomic Theory' (page 46) is easier to follow.