Enumerator for Word and Halting Problem
Let $\Sigma = \{0, 1\}$. Clearly $\Sigma^*$ is enumerable. For the word problem you can proceed as follows.

For each pair $(i,j) \in \mathbb{N}_+^2$ in dovetail fashion:

Let $w_i$ be the $i$-th word in $\Sigma^*$.
Check whether $w_i$ encodes a valid Turing machine $T$ (w.r.t. some fixed encoding). If not skip to the next iteration.
Simulate the Turing machine $T$ with input $w$ for at most $j$ steps.
If $T$ halts at the end of the $j$-th step, output $T$.



For the Halting problem you can do as follows:

For each pair $(i, j, k) \in \mathbb{N}_+^3$ in dovetail fashion:

Let $w_i$ be the $i$-th word in $\Sigma^*$.
Let $w_j$ be the $j$-th word in $\Sigma^*$.
Check whether $w_i$ encodes a valid Turing machine $T$ (w.r.t. some fixed encoding). If not skip to the next iteration.
Simulate the Turing machine $T$ with input $w_j$ for at most $k$ steps.
If $T$ halts at the end of the $k$-th step, output $\langle T, w_j \rangle$.