Learn French: Il parle d'une façon très calme.

Yes, you absolutely can! Il parle très calmement (He speaks very calmly) means exactly the same thing. Using d'une façon (in a way) or d'une manière (in a manner) followed by an adjective is a very common alternative to using an -ment adverb in French.

Why does the sentence use d' (from de) instead of a word for "in" like dans?

In English, we say "in a way", but French uses the preposition de (of/from), which becomes d' before a vowel sound. For these expressions, you will always use de, making it d'une façon or d'une manière. You cannot translate the English "in" literally here.

Is the adjective calme in its feminine form here to match façon?

Yes. Because façon is a feminine noun, any adjective describing it must be feminine. Since the masculine base form of calme already ends in an -e, the feminine form looks exactly the same. If you used an adjective that changes, like doux (gentle), you would see the feminine spelling: d'une façon très douce.

Could I use the word manière instead of façon?