My 1000/Linux can run Eclipse, but it only runs well when in high performance mode which shaves the battery life significantly.
Even the most ludicrous text editor you can think of should be fine. It wasn't that long ago that a 1.6ghz machine with 1gb of ram was top of the range.
Yes, it's fine for coding. I have an eee 901. I use xmonad and very small fonts and everything is great. It's really a wonderful laptop for traveling -- the battery lasts long enough for me to use it while waiting in the airport, and for the entire flight (most of the time).
The CPU is plenty powerful, too. One of my current projects is a 3d game, and it runs fine on the eee.
Same for linux: for the first time in my life, a non-geek opened the computer box, switched on, setup the wireless and connected to me via skype in a matter of minutes, while chatting on the phone about other matters. [edit: with video, and voip without headset and free of voice speaker to microphone coupling]
With the eee, desktop linux for everyone is a fact.
[And 7 hours of battery--they claim 8, but 7 is also great.]
"... A 'top' reveals there is no swap to speak of: ... Is that what you mean? ..."
Yes. This is with the Lin version so I wonder what they (Ausus) are doing not requiring a swap. It makes me think I'll have to look harder at the Ubuntu eee version to see if they swap if I get a eeepc.