Is "running" a verb or an adjective?
It can be either a verb or an adjective or even a noun. See explanation for examples.
The gerund form is a form of a verb, hoever it can have one of many functions in a sentence:
Noun (gerund): Running is a good thing for your health. Here the word is a subject therfore it functions as a noun.
Verb: A man is running to catch a bus. Here the word is a verb.
Adjective (participle): I saw a running woman in a park. Here the word is an adjective describing the noun woman.