So since Azrael is the angel of death, where does he take the deceased spirits? Does he have his own little corner of Heaven where he looks after the deceased or what?
It depends actually. Most souls will get reincarnated into a new life. Ophanim (thrones) act like portals from the Heavens to Earth. Even the Heavens have limited space. It's big bit still not enternal. It's just another realm.
When Azreal comes to pick up the departedsouls, he first comforts them before taking them to the Heavens.
All angels do have their own section in Elohim's region of the Heavens.
He was once one half of the first angel known as Helel. As a kid, he was very close to his twin brother, Lucifer. During the beginning of Lucifer's rebellion, Michael desperately tried to talk him out of it, to no avail. He later defeated Lucifer, ending the uprising. For a while after the war, he and Lucifer disliked one another. Due to being one entity long ago, they possess the urge to stay together. This caused Michael to seek out a relationship with his brother, asking Lucifer for a secret truce. Lucifer agrees and since then they often spend time together.
His current job is leading his father's, Elohim, army. To his subordinates, he keeps a stern front but on the inside, he stresses out constantly and is anxious. While he's not on duty, he's more relaxed and shows a kinder side. He'll often sneak out to take mini-vacations from the stress by visiting Lucifer.
He enjoys watching humans, fascinated by their ideologies and various cultures. At times he can get envious towards them, wishing his life was as simple and free as theirs. He's, also, jealous of their ignorance. His main concern is making his father proud and will do what he can to gain his father's approval.