What do you call words that share the same root (e.g., "network", "networks", "networking"). Also, does the shortest one of them have a specific name (e.g., "network")?
I would call network a "stem", networks (noun or verb) an "inflected form", networking (participle) an "inflected form", and networking (noun) a "derived term".
"Root" is not the same as "stem" in my book; network has two roots, net and work.
A set of words that can be grouped together is called a word family. A word family can be form-based or meaning-based. The question describes a form-based word family and the shortest form is sometimes a root, however, I agree with RegDwight's answer that the word network is composed of two roots.
Form-based families are important because they reveal sometimes hidden patterns of spelling in words that children already know; for example, the verb root pronounced 'seev' is spelt ceive (receive, deceive, conceive), and always corresponds to ception in the corresponding noun (reception, deception, conception).
Meaning-based families are important because they reveal links and patterns of meaning in words that children already know; for example, many adjectives and nouns are related as in the trio big - little - size. The specific meaning relations they contain (see below) are also an important component of reasoning skills.
In linguistics, words that share a common etymology are called cognates. Merriam-Webster gives this definition:
3a : related by descent from the same ancestral language
Cognates can be in the same language or two different languages. Wiktionary provides these examples:
English mother is cognate with Greek μητέρα (mitéra), German Mutter, Russian мать (matʹ) and Persian مادر (madar).
English shirt is cognate with English skirt, short, and curt ... all of these are descended from the Proto-Indo-European root (s)ker-, meaning ‘to cut’.