Not much is known about his life, mostly because of the existence of only a single biographical source and an abundance of legends.
John studied music, mathematics, and geometry, as well as theology--both Christian and Muslim. He was well-versed in the Qur'an (due to his submersion in an Arab culture), mostly so he could criticize it. He also wrote many hymns that are still in use today.
John of Damascus is most well-known for his defense of the icon. His life coincided with the rise of iconoclasm--the destruction of icons and images in the church because of the fear of idolatry. Leo the Isaurian of the Eastern Church first issued a decree against the veneration of images, and then one against their display entirely. Because of the Church's long tradition of venerating icons, John fought back, defending the importance of the icon in worship. He explained that it's not the icon itself that is worshiped, but that God can be more fully worshiped through the viewing of the icon:
"I do not worship matter, I worship the God of matter, who became matter for my sake and deigned to inhabit matter, who worked out my salvation through matter. I will not cease from honoring that matter which works for my salvation. I venerate it, though not as God."John's theology of the icon has many implications for incarnational theology. God created matter; God became matter in Jesus Christ, therefore matter is not itself evil, and can actually be used to reveal God.
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While I certainly wouldn't call John of Damascus egalitarian, he did hold Mary in high regard and important in the salvation story. So that's something.
I think that Damascene's appreciation for matter, that is--the physical world, as a result of God's creation is worth noting here. His veneration for icons I think could be carried over to a love for general revelation in nature, and therefore a certain care for the environment.
It's hard to say on this one. While John was often accused of being a worshiper of images and an idolater, he also fought against the Nestorian heresy and other unorthodoxy. But his trinitarian understanding was lacking. Also his eucharistic doctrine is a little wonky. He toes that heresy line pretty well.
Polymaths always get stars for badassery! Plus, the Byzantine emperor forged a letter in his name, saying that he was part of a plot against Damascus, and so the caliph cut off his hand! AND THEN BY A MIRACLE OF THE VIRGIN MARY IT GREW BACK! OK, that's probably just a legend but still.