Is walking to and from the the mosque a prerequisite for attaining the virtue of "He who washes on Friday and thoroughly cleanses himself, and comes early and is among the first to arrive..."? And does taking the elevator conflict with the condition of walking?

The virtue of walking on Friday to attain the reward mentioned in the noble Hadith (definition) is contingent upon fulfilling all the mentioned conditions, including walking and not riding. Using an elevator falls under riding, which negates the recommended act mentioned in the Hadith, as it prevents walking a portion of the path. However, upon returning, the worshipper has the choice to walk or ride, as the act of worship has concluded.