Need an Introduction to Tiny Core? Or peruse Core Concepts.
In .xsession, before jwm, without "exec". For example:opera -fullscreen
This file is saved in a backup, or could be added to an extension, since they are loaded before X starts.
tar cvzf myconfig.tce /home/tc/.xsession
Edit: for the timing problem, you could create a script that would sleep some seconds and then launch opera. This would launch it after Jwm, so it positions right like it should.