It's not really a matter of omitting an article. Grammatically, when you have one article, it's because there is just one noun phrase and a single article which comes before a single noun. This is because of the way conjunction works in language. When you combine several words or phrases with "and" or "or", you wind up with a single phrase of the same category as each of the things which were combined.
In your example "I will bring a pen, eraser, or pencil", when the three nouns "pen", "eraser", "pencil" are combined with "or", you get a noun (not a desk set), because the things you started with are nouns. Diagramming the structure, we have
[NP a [N [N pen] [N eraser] or [N pencil] ]
That is, "pen eraser or pencil" is a noun, so naturally it is preceded by a single article "a".
Alternatively, the same thing could be expressed with a NP which is a conjunction of three NPs, in which you naturally wind up with three articles, because each NP gets its own article:
[NP [NP a [N pen]] [NP an [N eraser]] or [NP a [N pencil]] ]
Similarly, for your example 3, there is a choice between using a single adjective formed by conjoining the three adjectives "black", "yellow", and "red", or a single noun formed by conjoining the three modified nouns "black shirt", "yellow shirt", and "red shirt".
Your example 4 is more complicated, since it is derived the preceding by Right Node Raising (RNR for short) by moving a single constituent shared by all conjuncts up higher in the structure.
I will select [NP [NP a black,] [NP a yellow,] or [NP a red,] shirt].
"shirt" is the node that is raised. It may be preceded by a comma, marking the intonation break that is heard in such constructions. (The analysis of RNR constructions is controversial.)