Is it true that my RMI applet can make socket connections only to the host from which the applet was downloaded from?

Yes, the default behaviour for applets (including those that may be RMI clients) is to communicate with an RMI server that is hosted on the same platform from which the applet was served from.

It is one of the manifestations of the applet sandbox paradigm, and can be overcome by deploying "signed applets" which can go beyond the sandbox.