He's not bound to time, is what's happening, I think. Babylonia is the earliest singularity in the timeline and so he can manifest himself there because it's not affected by the condition that he cannot leave Avalon after his time in Camelot, or something.
Astolfo using Kosmos at a downgraded state needs to be proven with sources.
In the light novel, Achilles doesn't use it himself, gave it over to Astolfo for him to use. Naturally, transferring Noble Phantasms would normally never happen in a regular Holy Grail War. Even in the Great Holy Grail War, it would normally be unthinkable.
In the first place, most Noble Phantasms are tied to the legends of heroes. Even if you were to borrow Gae Bolg from the blue spearman, that doesn't mean you would be able to activate Gae Bolg. However, there are exceptions. In this case, in addition to the necessary conditions of "being where he is making a contract, instead of being in opposition, of his own volition" and the Noble Phantasm "not requiring considerable skill and power to invoke its true name", it is believed that the transfer of the Noble Phantasm went smoothly because there is "an episode in the legend of the giver (Achilles) where he loans his Noble Phantasms" and "an episode in the legend of the receiver (Astolfo) where he borrows a Noble Phantasm". - Fate Apocrypha Material Encyclopedia translated by NanoDesu
That said, I do agree with Kosmos being Greece instead of the world itself. I mean, look at the description:
The world that Achilles saw. The outer circumference portion has a swirling ocean currents because of the sea god.
- Fate Apocrypha Materials Status
As far as I know, Achilles never left Greece.
