1. Je vais vous montrer quelques chiens. Voici mon plus rapide.

   That is correct; but you could express that better:

   Je vais vous montrer quelques uns de mes chiens: en voici le plus rapide

2. Je vais vous montrer quelques chiens. Voici un qui est rapide.

   That is almost correct:
   Je vais vous montrer quelques chiens: en voici un qui est rapide.

3. J’ai deux chiens. Voici mon rapide.

   That’s incorrect. Here you have the personal pronoun Je so using the possessive pronoun mon in this context is rather a bad option. I would prefer to express your idea this way:

   • J’ai deux chien: en voici le plus rapide.
   • J’ai deux chien: voici le plus rapide.

4. J’ai deux grand-pères. Voici mon paternel.

   That’s incorrect. We don’t say that in French. You can express your idea this way, for instance:

   • J’ai deux grand-pères: celui-ci est de mon coté paternel.
   • J’ai deux grand-pères: voici mon grand-père paternel.

I was not sure whether this qualifies as an answer as I am not able to provide any logic to explain it but I decided to post this because my intuitions about this are very different from Begueradj’s.

Je vais vous montrer quelques chiens. Voici mon plus rapide.

Sounds OK, even if “le plus rapide”, which you mentioned as well, feels more natural.

Je vais vous montrer quelques chiens. Voici un qui est rapide.

Does not sound natural, I would say “En voici un qui est rapide” instead.

J’ai deux chiens. Voici mon rapide.

That does not sound good to me. “le plus rapide”, “le plus rapide des deux”, “le plus rapide d’entre eux” or “le plus rapide de mes chiens” would all work but are obviously quite different.

J’ai deux grand-pères. Voici mon paternel.

Also sounds odd to me. I think I would simply repeat “grand-père” as in “Voici mon grand-père paternel”.

Note that both “parternel” and “rapide” have other meanings that collide with the adjectival use and might make your examples a bit confusing. “Mon paternel” means “my father” is (a somewhat old-fashioned) slang and “le rapide” was used to designate a fast train (also a bit old-fashioned).