Learn Russian: Он говорит по-русски довольно хорошо.

No, you cannot. Гораздо and намного (meaning 'much' or 'by a lot') are used specifically to intensify comparative words, like лучше (better). Because хорошо is in its basic, non-comparative form ('well'), you must use words like довольно (quite) or очень (very) instead.

Why do we use по-русски here instead of just русский?

The verb говорить (to speak) requires an adverb to describe how an action is done. The prefix по- and the ending -и turn the word into an adverb, meaning 'in a Russian way'. If you wanted to use русский, you would need the full noun phrase as a direct object: Он знает русский язык (He knows the Russian language).

I have seen довольно used to mean 'enough'. How do I know when it means 'quite'?

It depends on where it is placed. When used directly before an adjective or adverb (like хорошо here), it acts as a degree modifier meaning 'quite' or 'rather'. When used on its own, especially as an exclamation (Довольно!), it means 'Enough!'.