You launch a Terminal session by clicking on K Menu and selecting the appropriate entry in the menu. I'm on Ubuntu at the moment, so can't tell you the precise words. When the Terminal window is open, you would just type firefox at the prompt (or maybe it's mozilla-firefox). The Firefox browser window would then open and the Terminal window will stay open. When Firefox crashes you may get some messages appearing in the Terminal window. I assume you're running the 32-bit version of Firefox, as there is not even a Sun Java plug-in for the 64-bit version of Firefox?
Regarding setting up the JRE, search for "Sun" in Kuroo or Portato and you should see the Sun JDK (or whatever it's called) which includes the JRE. Just emerge that. Then configure Konqueror to point to the Sun Java binary:
1. Launch Konqueror.
2. Settings > Configure Konqueror...
4. In the box next to "Path to Java executable, or 'Java':" enter something like the following (will depend on what you just emerged, so have a look in the /opt/ directory to find it):
Can't give you the precise path in Step 4 because I'm on an Ubuntu PC at the moment. But it works fine for me using SL x86-64.
EDIT: Don't be fooled by the example I gave above into thinking that you'll necessarily find "sun-jre-6u...", as I installed the JRE outside of Portage many moons ago before the ebuild for the latest JDK was in Portage and I just pulled the above example from an old post of mine in the SL Forum. It might be "sun-jdk-something or other". The last time I did this, I did it the proper way, i.e. emerged the JDK from Portage and entered the path to the binary in Konqueror as explained above.
EDIT2: Just found another old post of mine which gives the path the last time I did it:viewtopic.php?f=5&t=11008&p=63669&hilit=firefox+konqueror+sun+java#p64241