Who takes our lives?
In Quran we find three different and apparently contradictory statements:
- One saying it is the angel of death in surat as-sajdah 32:11 ... note that this verse can be understood that for each one of us there's such an entrusted angel. However most tafsirs i consulted say it is one angel.
- An other saying that those are many angels like in surat an-Nisa' 4:97, surat al-An'aam 6:61 and surat Muhammad 47:27.
- And a third statement saying it is Allah in surat az-Zumar 39:42.
these differences can be explained as follows assuming there's one angel of death we can conclude from the other verses that this angel only acts by Allah's order so it is by his order that he takes a life (ruh or rooh) and on the other hand this angel might have helpers among angels, as he as the angel of death is the angel that is ordered to receive a rooh once the person has died. This would also be an explication if we take into account my note that every person might have an angel which is entrusted to take the rooh/soul, but Allah knows best!
Do angels die?
According to Quran everything dies or has an end expect Allah as you may read in surat al-Qasas 28:88 or ar-Rahman 57:26-27. Note that the later verse can't be used as a proof for the malaika as it quotes those creatures that are "on Earth".
There's a munkar hadith (this means a rejected hadith) which you may read in the Arabic original here which is partly also quoted in tafsir at-Tabari of verse 39:68 which describes the end of the world and the day of resurrection or as al-Maududi says:
This is a figurative way of describing the complete control and authority of Allah over the earth and heavens. Just as a man encloses a small ball in the hollow of his hand with perfect ease, or a person rolls up an handkerchief in his hand without any difficulty, so will All men (who fail to conceive the greatness and glory of Allah) sec with their own eyes, on the Day of Resurrection, that the earth and the heavens arc like an ordinary ball and a small scroll in the hand of Allah. ...
Note that according to this munkar hadith the angel of death will take the lives of the few angels who still have not yet died until there's only him and Allah and than Allah will say to him: die and he would die.
Also note that ibn Hazm and some other scholars say that angels won't die based on this verse of surat az-Zumar 39:68 as it clearly says.
And the Horn will be blown, and whoever is in the heavens and whoever is on the earth will fall dead except whom Allah wills. ...
Mufassireen like at-Tabari concluded that angels are among those who will stay alive, so there might be creatures which may stay alive! This finds also some support in surat al-A'raaf 7:20:
He said, "Your Lord did not forbid you this tree except that you become angels or become of the immortal."
which may support the view that angels wouldn't die! An objection to this could be the fact that angels have been quoted separately from the immortal which may mean that they are not among the immortal creatures.
Finally: There's no reliable hadith saying that the angel of death will get itself (or himself). And there's no clear Quranic verse which may even support the idea that angels would die at all, but there are interpretation of verses from which one could conclude this and others which oppose this idea.
See also this fatwa.