I have two questions:
Can I use applet to install APK applications?
Is it possible to send and install an APK application using OTA?
Summary, can I send an applet that contains an APK and install that?
AFAIK there is no way how an (U)SIM card applet could directly install an APK.
But it might be possible if there was an agent application installed (in the phone) which would install the APK on behalf of the (U)SIM applet (of course this application would have to have permissions to install APKs).
This agent application could communicate with the applet using the SIMalliance Open Mobile API (as long as the phone supports this API).
(Being enlightened by your other question)
Maybe you could trigger APK installation by opening the browser with the URL pointing to the APK file.
Good luck!