= Solution
By part (c), $T$ has a <first-order model>, so it is consistent. By part (d), every <first-order sentence> is equivalent modulo $T$ to a <quantifier-free sentence>. In the <first-order signature> consisting only of $E$ and <first-order equality>, there are no constant symbols, no ground <first-order terms>, and no nullary predicates. Thus a closed <quantifier-free formula> reduces to true or false, using the logical truth constants. All <first-order models> of $T$ therefore agree on every <first-order sentence>:
$$
\boxed{T\text{ is complete}.}
$$
The construction in part (d) is effective. Conversion to <disjunctive normal form>, enumeration of finite equality partitions and adjacency patterns, and testing the finitely many witness cases are all terminating operations. Apply them recursively to a given <first-order sentence> and evaluate the final truth constant. This decides whether the <first-order sentence> belongs to $T$, giving
$$
\boxed{T\text{ is decidable}.}
$$
No useful bound on the running time is needed for decidability; the potentially large enumeration still always terminates.
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