1. None of his powers are innate, he would be unable to cast magic without his connection to the Light/Shadow. He's considered to have a special affinity for the Light but it comes to him like a shaman summoning elemental spirits. There are circumstances which could sever his connection to it. The oracle is obscure. They haven't explained what powers created this condition in him, and it has no parallel in the lore. It is innate in the sense that it is a part of his body, but I don't know if it would fail if the Light could no longer reach him.
2. He can restore the recently dead to life, but it is extremely uncommon- he has only accomplished it twice, once with the help of another priest and a divine being. The one time he resurrected someone by himself, it was an act of extreme desperation so he did it more or less intuitively. I assume that should he attempt to save someone intentionally, he would have a much greater chance of failure. As for his general healing, he gets 'tired' easily. Any situation where he was needed to heal for extended periods would quickly exhaust his powers and he would need to withdraw and rest.
3. The shields will prevent a certain amount of incoming damage but have a finite duration. They don't dissolve once a certain amount of damage has been taken, but simply wear off. The duration is very short- just a matter of minutes. There is a shield applied to a single person, and a dome-like AOE shield. These are all taxing spells which he cannot use repeatedly in quick succession. Visually, they look like a golden bubble.
4. As far as I know, there is no special power inherent in the sword other than it has the ability to be split into two blades and melded back into one again at will. It glows to show a soul-bond with those who are attuned to it. It was his father's. When he died, it lost any special appearance until Anduin picked it up and attuned himself to it. Its ability to split may have been connected to his father. Basically, it may just be really shiny at this point :D
ETA: This is the only evidence I have that the sword is still 'magic' in Anduin's keeping. I don't know if it's just 'rule of cool' here, or if this is a power the sword has. It looks like an in-game power called 'Wake of Ashes' for a different lore sword. That power was an infrequent effect where enemies (in that case, undead) were turned to ashes. https://youtu.be/jSJr3dXZfcg?t=2m55s
5. There is a clairvoyant aspect to it. He was able to ask it about someone's motives, for example. But it also will cause him pain if he is about to do something which will harm innocent people (or even if his inaction will cause harm- such as when someone was about to attack people he had promised to protect). Canon is unclear if the oracle is connected to the powers of his world, though.
Re: QUESTIONS
2. He can restore the recently dead to life, but it is extremely uncommon- he has only accomplished it twice, once with the help of another priest and a divine being. The one time he resurrected someone by himself, it was an act of extreme desperation so he did it more or less intuitively. I assume that should he attempt to save someone intentionally, he would have a much greater chance of failure. As for his general healing, he gets 'tired' easily. Any situation where he was needed to heal for extended periods would quickly exhaust his powers and he would need to withdraw and rest.
3. The shields will prevent a certain amount of incoming damage but have a finite duration. They don't dissolve once a certain amount of damage has been taken, but simply wear off. The duration is very short- just a matter of minutes. There is a shield applied to a single person, and a dome-like AOE shield. These are all taxing spells which he cannot use repeatedly in quick succession. Visually, they look like a golden bubble.
4. As far as I know, there is no special power inherent in the sword other than it has the ability to be split into two blades and melded back into one again at will. It glows to show a soul-bond with those who are attuned to it. It was his father's. When he died, it lost any special appearance until Anduin picked it up and attuned himself to it. Its ability to split may have been connected to his father. Basically, it may just be really shiny at this point :D
ETA: This is the only evidence I have that the sword is still 'magic' in Anduin's keeping. I don't know if it's just 'rule of cool' here, or if this is a power the sword has. It looks like an in-game power called 'Wake of Ashes' for a different lore sword. That power was an infrequent effect where enemies (in that case, undead) were turned to ashes. https://youtu.be/jSJr3dXZfcg?t=2m55s
5. There is a clairvoyant aspect to it. He was able to ask it about someone's motives, for example. But it also will cause him pain if he is about to do something which will harm innocent people (or even if his inaction will cause harm- such as when someone was about to attack people he had promised to protect). Canon is unclear if the oracle is connected to the powers of his world, though.