How is a deceased person considered a martyr in Islam, and is it permissible for a person to ask God to let them die as a martyr and not live to a decrepit old age?

Martyrdom (shahada) in Islam comes in various forms. Supplicating for martyrdom is legitimate, as Umar ibn al-Khattab supplicated for it, and Al-Bukhari titled a chapter "Chapter on Supplicating for Jihad and Martyrdom for Men and Women."

Supplicating not to be returned to the most decrepit age (ardhal al-umur) is legitimate, and the Prophet, peace and blessings be upon him, sought refuge from it. Among the Prophetic supplications in this regard are: "O Allah, I seek refuge in You from cowardice, and I seek refuge in You from being returned to the most decrepit age, and I seek refuge in You from the trials of this world, and I seek refuge in You from the punishment of the grave," and also "O Allah, I seek refuge in You from laziness, and I seek refuge in You from cowardice, and I seek refuge in You from old age (haram), and I seek refuge in You from stinginess."