Let $\mathcal{B}$ be the Borel σ-algebra on the Euclidean metric space of $\mathbb{R}$. For any non-empty set $A \in \mathcal{B}$, is it always true that $A$ can be written as a countable union of bounded intervals in the following forms: $(a, b)$, $[a, b]$, $[a, b)$ and $(a, b]$?
For example, we can easily write $(-\infty, 4) \cup [5, 6) \in \mathcal{B}$ as $$(0, 4) \cup [5, 6) \cup \bigcup_{i = 1}^{\infty}[-i, 0] $$
In other words, is it possible to construct (not generate) the whole set of $\mathcal{B}$ with solely countable unions of bounded intervals? (And include the empty set into $\mathcal{B}$ as well)
This is most certainly not true. As a counterexample, consider the Cantor set (https://en.wikipedia.org/wiki/Cantor_set). Firstly, the Cantor set contains no intervals of positive measure and so the only sets we can use to construct it (as per your restrictions) are singletons $\{a\}, a \in \mathbb{R}$. However, the Cantor set is uncountable and therefore cannot be written as the countable union of singletons.