A lot of questions and some confusion. I'll try a few.
-time flowing normally in expansion: that's only because the time that is tied to the expansion flow, is the time in the rest frame of the bodies that are flowing with the flow. We are them. It's called cosmic time because anywhere in the universe it applies the same way. It is a result and artifact of the uniform and isotropic universe and flow.
-in a gravity field or to be more exact in a region where the gravity is stronger, like near the Sun, or near a Black Hole's horizon, it slows down, compared to where gravity is weaker. On the surface of the earth it goes slower than a few hundred or thousand miles up. For instance it is faster at the GPS satellite positions than on the surface of the earth, by a very small amount, but clocks need to be adjusted for it.
-out in interstellar space there are regions where there are more galaxies (really a higher mass density), than in some other regions (called voids, they are no real voids just lower mass density). This happens statistically due to normal variations. In the voids the clocks go faster, by the corresponding amount, than near a lot of galaxies.
-how time flows depends on gravity. The speed of light does not. The way light is affected is it undergoes a redshift if it is coming out of a gravity well (high gravity), and blue shifted if going in. It can also be deflected (bent) by strong gravity (eg when it comes close to the Sun, it bends in, same going around as it skirts a Black Hole). But the speed is unchanged, measured by a local observer
-space can only expand slower if there was more mass density on the average in the universe. As it is, there is less than the dark energy, which makes it expand faster (very slightly now, but it is accelerating). Speed of light does not change.
-Oh, and the universe expansion basically does not affect the gravity or curvature near a Black Hole horizon. The Black Hole gravity is just too strong. Now, to be sure, that's now, anywhere in the universe, but as it expands faster it can have more of an effect. It'll about hundreds of billions of years tear up all the matter, probably only Black Holes will remain at late times, eventually they will all evaporate and as the universe continued accelerating and expanding it'll be mostly void.
-See the wiki article on the speed of light at https://en.m.wikipedia.org/wiki/Speed_of_light
See the wiki article on accelerated expansion at https://en.m.wikipedia.org/wiki/Accelerating_expansion_of_the_universe
Hope this clear up some confusion.