I discovered what looks like a great free tool for developing Flash components and web sites, called OpenLaszlo. Based on my very cursory investigation, applications can be developed using just a text editor, but they create very nice Flash output. An eclipse plugin is also available from IBM.
The OpenLaszlo SDK uses a servlet, which I couldn’t run at my current host, but I discovered EATJ, which offers free basic Tomcat hosting. Laszlo Systems also has a Developer Zone, with forums and free non-commercial hosting at mylaszlo.com.