XML-RPC does the trick, IDEA maybe
What's the best XML-RPC implementation for Java at the moment?
Apache XML-RPC (originally Helma) - http://xml.apache.org/xmlrpc/
For a simple API is XML-RPC a better bet than SOAP?
No - build it using Glue, it's SOAP made easy for Java. It's a dream to use. And free for most uses.
Is it safe to use Swing in applets these days?
Wrong person to ask. 
What are the relative advantages/disadvanages of JWS over Applets? Maybe even thinlets?
Complex question. Thinlets are only really good for small tasks, but are very rapid to develop. JWS is easy to write and 'deploy' any Swing app anywhere. I'd say JWS Swing is better than applets, but I'm biased.
And what IDE should I be using (I used to use JBuilder)
Easy - IDEA.
ยป Okay I'm finally catching up on all the news I haven't read for a few days. Some interesting points from Mike.
As luck would have it Apache XML-RPC is the one I ended up downloading. I had it hooked up to Radio and exchanging data in about as long as it took me to type this sentence, that was pretty cool.
I also choose to re-evaluate IDEA. I actually came across this IDE a long time ago when I was using some of their refactoring tools for JBuilder. At the time I dismissed it because (a) it didn't do very much that the free JRefactory tool couldn't already do, and, (b) I was doing GUI stuff and JBuilder does that.
I like a lot of what I see in IDEA and it's certainly a lot cheaper than JBuilder. I just wonder how I'll hack the GUI bits and pieces without a visual designer...