First of all, eggs can be traded, for sure.
Second of all, be sure to familiarize yourself with the information in this guide. Very near the beginning answers your question about the Pal Park.
Now then. The hometown is the game in which the egg was obtained, not hatched.
The egg received at info is also retained (bearing in mind that this will show as "Faraway Place" if traded from a new location in Platinum or HGSS; the location 2004, "Johto Region", is not used at all).
I'm not sure, as HGSS is only available in Japanese, but I believe HGSS retains the location data from DPPt, so trading a Pokémon from DPPt to HGGS, the received at data should remain the same. In other words, if you got your egg from Riley in DPPt, it should still say Riley in HGSS.
So basically the rules of trading between versions works the same way with eggs as it does normal trading; the information basically doesn't change, just which information is displayed on the Trainer Memo might change.