Does gravity slow the speed that light travels? Does gravity slow the speed that light travels?  Can we actual measure the time it takes light from the sun to reach us?  Is that light delayed as it climbs out of the sun's gravity well?
 A: They call it relativity for a reason.  The speed you measure light traveling depends on your reference frame when a gravitational field is present.  If you are in the rest frame of the mass that is generating the gravitational field, then the speed of light can be calculated from the Swarzschild metric. This link presents a nice derivation:  http://mathpages.com/rr/s6-01/6-01.htm
In general, the speed of light depends on the gravitational field as well as the where and what direction the beam of light is.  The speed of light is not a simple scalar in a gravitational field, but rather a tensor!
A: This is one of those questions that is more subtle than it seems. In GR the velocity of light is only locally equal to $c$, and we (approximately) Schwarzschild observers do see the speed of light change as light moves to or away from a black hole (or any gravity well). Famously, the speed that radially moving light travels falls to zero at the event horizon. So the answer to your first question is that yes gravity does slow the light reaching us from the Sun.
To be more precise about this, we can measure the Schwarzschild radius $r$ by measuring the circumference of a circular orbit round the Sun and dividing by 2$\pi$. We can also measure the circumference of the Sun and calculate its radius, and from these values calculate the distance from our position to the Sun's surface. If we do this we'll find the average speed of light over this distance is less than $c$.
However suppose we measured the distance to the Sun's surface with a (long) tape measure. We'd get a value bigger than the one calculated in the paragraph above, and if we use this distance to calculate the speed of the light from the Sun we'd get an average speed of $c$.
So I suppose the only accurate answer to your question is: it depends.
Re your other question, assuming the spacetime around the Sun is described by the Schwarzschild metric, the time dilation at the surface of the Sun is given by:
$$ \text{time dilation factor} = \frac{1}{\sqrt{1 - r_s/r}} $$
where $r_s$ is the radius of a black hole with the mass of the Sun and $r$ is the radius of the Sun. The former is about 3,000m and the latter about 700,000,000m so I calculate the time dilation factor to be around 1.000002 and this is too small to measure directly. 
However you can interpret gravitational lensing to be due to changes in the speed of light, and since we can measure the gravitational lensing due to the Sun you can argue we have measured its effect on the speed of light. This isn't really true as what gravitational lensing measures is the spacetime curvature. However the change in the speed of light (measured by a Schwarzschild observer) is an aspect of this.
A: 
Does gravity slow the speed that light travels?

Not really. Light just follows a curved path in the curvature of space-time produced by a massive object (a consequence of gravity). But, gravity itself doesn't slow down light. Because, we've just corrected these gravitational waves (a century ago) to not to be instantaneous, but travel exactly at $c$ as a consequence of SR, which declares speed of light to be a local frame constant. So, it doesn't arise to slow down light.
But, it depends on how you measure. If you're an observer and you measure $c$ locally anywhere, you'll be able to say that it's still a constant. Locally, it's a No... But, if you're looking towards a massive object like a black-hole or a neutron star, you can measure light much slower or faster depending on where you are. I mean, whether you're influenced by the fields very strongly.

Can we actually measure the time it takes light from the sun to reach us?

Yes. But, we've to take the gravitational time-dilation into account.

Is that light delayed as it climbs out of the sun's gravity well?

Yes. As we measure this light taking geodesic paths around these gravitational fields, there's this Shapiro delay. But, as we take the appropriate measurements of distance and time along the geodesic, we can still find that $c$ is a constant. 
A: *

*No, Maxwell's equations mean that's constant. 

*Yes, it's ~8 mins. 

*Yes, if it comes on a curved path. For example some (but not a lot for something as light as the sun) will leave the sun heading away from the earth but will swing back around as it's attracted to the sun. The total distance traveled is longer than a straight line path and this will be reflected in the time taken.


See also: gravitational lensing.
