Examples of infinite sets of regular and non-regular languages that their union is regular and non-regular
a. is pretty easy. Just consider $\{a\}^\ast = \bigcup_{i \in \mathbb{N}_0} \{a\}^i$. This also answers the last question in your post (i.e., regular does not mean finite).

The answer to b., as you have said, can be found in the linked question.

Finally, for c. and d. you can use subsets of $\{ 0^n 1^n \mid n \in \mathbb{N}_0 \}$ and $\{ 1^n 0^n \mid n \in \mathbb{N}_0 \}$, which are both non-regular languages. Hint: The exercise text does not require the languages to be disjoint.