Most of your questions were answered pretty well already, but there are few points that were left out.
For the phonton time is as normal, without running slowly.
That is not so. Relativity cannot give any sense to concept of time or space for the photon. In relativity the notion of space and time are relative, they are just different view on one entity that is spacetime. The spacetime is the mathematical object that describes the "stage of our universe" on which things are happening. We, as subluminal observers, can (artificially, albeit the construction is physically meaningfull) separate spacetime on time and space, but photon can not. But that is nothing to be worried about, because separation to time and space is only, as i said, artificial.
But what happens when light travels away from the center of gravity of a heavy star? Does it slow down?
The gravity is described by general relativity. General relativity tells you, that the spacetime i was talking about is curved. The curved spacetime actually makes separation on time and space even more artificial.
The separation to time and space is done through the notion of simultaneity. That is two events that happened simultaneously you say happened at given time. Then space is set of all the events in the universe that are happening at the same time. But because what is called simultaneous depends on who is defining it, so is the separation on space and time observer dependent.
But in curved spacetime there is (in general) no natural way of defining simultaneity. Great examples are black holes. The distant observer is completly separated from the events happening inside the black hole. So how could he determine wheter his clock showing 9am happened at the same time as some unfortunate astronauts clock inside black hole? He has no way of knowing what is happening inside.
You can still separate universe on space and time, but this time it will be more and more just a mathematical trick the stronger the gravity is (and the further away from yourself you are trying to do the separation). This is of course very vague, but i think it conveys the main message.
That being said, the curved spacetime is not curved too strongly. I mean by that, that if you hide yourself inside small enaugh box, you will not see any curvature inside a box. That is similar to Earth - when you consider only small area (especially in the ocean) the Earth will be flat to high accuracy, even though the Earth is not flat at all. So no matter how strong the gravity is, you can always find box so small that the spacetime inside will appear flat with enaugh accuracy. Then general realivity will tell you that the physics inside your box will be same as if there was no gravity at all. So inside the box (and if it is in free fall you have basically inertial system), you can separate spacetime on space and time and measure velocity of light as distanced traveled divided by time passed. And since the physics inside the box is as if there is no gravity at all, the speed of light is always c no matter how strong of the gravity source there is outside of your small box.
So no, the observer inside the box will not see light slowing down.
But what happens if observer is not inside the box? This observer would also need to measure some distance that light traveled and time that passed and divide it to get speed of light. But as we said, the separation on what is space and what is time becomes more and more artificial the further away you are from the spot you are measuring and the stronger the gravity is. And this kind of speed does not need to be c at all. For example the distant observer looking at the light coming radially away from some heavy star and that is in rest with respect to this star, would see the speed of light to be (given by schwarschild metric):
where $c$ is the speed of light, $M$ is the mass of the star, $r$ is how far away from the star the light is and $G$ is gravitational constant. So yes, this observer would see the light slowing down, but that is only from his point of view. From the point of view of the observer that is right near the light it would still be c.