Should one who dies a martyr—such as one who drowns, is stabbed, or defends his honor—be washed before burial? And was 'Umar ibn al-Khattab, may Allah be pleased with him, washed?

The martyr of the battlefield is the only one who is not washed or prayed over, according to the majority of scholars. As for other martyrs, they are washed and prayed over. Umar ibn al-Khattab, may Allah be pleased with him, was washed and prayed over because he was carried from the place where he was stabbed and lived for a period thereafter, so he is not a battlefield martyr in the sense that would prohibit washing and praying over him. And "the one afflicted by the plague" refers to someone who died of the plague, not someone who was stabbed with a knife.