Did the expansion rate of the universe exceed lightspeed?

Yes, there most certainly is a relationship between time and space - this is the heart of Special Relativity. However, the relationship is based upon hyperbolic geometry, so at .999c, your time would be the square root of (1 - .999^2) as a fraction of normal, so about .045At .5c, your 4-velocity would be projected .86 into time and .86 into space. But an observer would simply see you travelling at .5c - if he looked closely, he would see your time having slowed, by watching your watch tick, or your heart beat. He would also see your length contracted in the direction of travel. Both of these are the result of the above space and time projections.There is no way to travel such that you age the same as the person who stays behind. The longest path through space-time is that that experiences no acceleration. Any acceleration will always shorten the path, and hence reduce your age between two events. Yes, it is very tempting to jump through time to see what happens - I'd certainly be tempted, though given the current state of our space-faring technology, I don't hold much hope for a near-c spacecraft in my lifetime! Alternatively, you can get the same effect by hanging around a black hole. Sadly, to 'hang' around a black hole requires propulsion systems that could easily take you to near-c.

If there were only the earth in the Universe, your Universe would be infinite in extent, and described by the Schwarzschild geometry (with a slight Kerr-Newman perturbation). In this case, lyx2no is correct - the stone would be travelling at around 11.2 km/s when it hit, assuming it was released from some sufficently large distance away, and ignoring air resistance. It is independent of exactly how far away it is dropped, and as space is infinte in this case, you can go as far away as you like.In an expanding Universe, a stone sufficiently far away will never make it to the Earth as the expansion will work to increase the distance between the stone and the Earth.

There are no objects 2.5 billion ly apart that will end up at the same point (neglecting re-collapse) - you are thinking of M31, the Andromeda galaxy, and The Milky Way. These are a mere 2.2 million ly apart - practically on top of each other from a large scale structure point-of-view. And a collison in 2.5 billion years means that they are moving at a small fraction of c with respect to each other.Neighbouring galaxies (such as ours and M31) are largely unaffected by the expansion of the Universe, becasue local gravity dominates. The expansion visibily operates on the scale of clusters and super-clusters of galaxies.

No, both would exhibit time dilation. You might differentiate them and call one relativistic time dialtion, and the other cosmological time dilation, but both are caused by identical phenomena - their time directions are rotated with respect to ours. In the former, the rotation is casued by the relativie velocity; the latter, by the actual curvature of space-time tipping over the distant light-cone with respect to ours.Edited by cavediver, : No reason given.

Sylas was addressing Tired Light, which assumes a static source,non-expanding space, and some mechanism to redden the light as it travels towards us. It cannot explain observed time dilation in observed objects, such as seen in th SN spectra that Sylas mentioned. However, it is very difficult to differentiate the relative contributions of cosmological and relativistic red-shift for a single object. That is where statistical analysis of large numbers of objects becomes valuable.Time dilation is the same regardless of the direction. Doppler is highly direction-dependent. Blue-shifts remain blue-shifts, although lessened, after time-dilation correction. Transverse
objects experience red-shift solely from time-dilation. Red-shifts are accentuated by time-dilation. So Arp still has to explain why all his galaxy-associated high-velocity quasars are all heading away from us!!Edited by cavediver, : No reason given.

I've just perused the fourteen companions of M31 listed on Wiki - seven are higher z than M31, five on lower (more negative z), and there is no data for two. Hmmm...