Importance of space constructability in time space relation in complexity


I am reading Arora-Barak’s Complexity book. In Chapter 4, they state and prove the following theorem.

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Why $ S$ should be space constructible? Wouldn’t all three containments of theorem hold, even if $ S$ is not space constructible?

My other question is about Remark 4.3, the book claims that if $ S$ is space constructible then you can make an $ NSPACE(S(n))$ machine halt on every sequence of non-deterministic choices by keeping a counter till $ 2^{O(S(n))}$ . I am not sure how we can keep such a counter in $ S(n)$ space. The space constructability of $ S$ implies that we can compute $ S(n)$ in $ O(S(n))$ space, not $ 2^{O(S(n))}$ in $ O(S(n))$ space.