Do any of the words in The Road Not Taken have connotations beyond their literal meanings that are significant in understanding the author's intentions?
Yes, some of the words in Frost's classic poem have connotations beyond the literal.
The most important by far is "the road." In many literary works throughout time, a road (or path, before there were roads) symbolized the path of a person's life. It symbolizes your fate, or how your choices unfold. You can see this in Dante's The Divine Comedy. The roads diverging are the two paths the narrator's life might have taken.
Another word that carries more meaning than the literal is "fair." Here the meanings come from the ambiguity of the word itself. It can mean just, or fitting, or beautiful. At this point the poem is saying that no matter which road (or choice) the speaker took, the path would be equally appropriate, and equally beautiful.
And finally, look at "black." Black has long been associated with both sin and stain (like wear). The leaves on both paths are untouched. They are innocent.