O.K. I've got everything together. PalmIIIe, Ordesktop, Omniremote and Remote Extender from Smarthome. The latter only took a week from order to delivery. Total cost without the Remote Extender was just over half that of the Pronto at $399. Nearly everything works as advertised and I am very pleased. Thanks to all the posters who figured this out and took the time to explain it.
The one exception is my Sony CDP-CX100 CD player. Nothing seems to make it work with the Palm. I've 3 other remotes that will make it operate through smoked glass. I have tried the Palm right up against the IR window of the CD player. The Palm/Remote Extender combo works all my other equipment I don't think it's the emitter of the Palm or the Remote Extender. I concluded that the IR codes must not be being captured correctly by the Palm. I've tried Pronto codes for various other Sony CD players, through Ordesktop, but to no avail. (There doesn't seem to be any Pronto codes for this particular remote.) So, please, does anyone have any knowledge or ideas of what I could try next.
