Is a soldier killed during duty by an armed group considered a martyr who is not to be washed, and if so, of which type of martyr is he?

The type of martyrdom depends on the cause of death. The martyr who is not washed is specifically one who died in war against disbelievers. As for other martyrs, they are washed. So, whoever is killed by an armed group, not in combat with disbelievers, is washed, shrouded, and prayed over like the rest of the Muslim dead.