We measure distance in space, and time intervals in time, and so velocity is just plain old distance/time. Special relativity doesn't change that. What changes is that if you start traveling at a different velocity, your measurements of distances and time intervals deviate.

The expansion rate of the universe is not a velocity in the usual sense of distance/time. It's actually in units of velocity/distance, which reduces to 1/time. An expansion rate of r Hertz means that a given span of distance intrinsically doubles every 1/r seconds. The objects occupying the space don't "move" in any real sense due to expansion. They just wind up farther apart because space itself grew.

And, just like measurements of distance and time, measurements of the expansion rate change if you change your velocity. There is a special velocity in our universe which causes the expansion in all directions to be the same. From this special perspective, which is traveling at a kind of cosmic "rest" velocity, you can calculate the expansion rate. It turns out that the Sun is traveling at approximately 370 km/s with respect to that special "rest" velocity.

Yes, it is the same thing, but since the objects are in free fall and there is no traditional force to cause the acceleration the better view point is that this is accelerated expansion of the universe. In a flat spacetime a forward light-cone can be identified with an expanding (no acceleration) universe where objects just fly away from a single point with constant but different speeds, i.e. an explosion. But in this model space as slice with the same local time after explosion is not flat. Also data seems to indicate that space is flat while space-time is curved on a large scale, so this picture is too simple.