I have experienced some problems with Firefox and flash, in that it takes up quite a lot of system resources (CPU time mainly) when they are running. This is especially true of eBay as it contains quite a few flash movies.
As for Java, the only time I have problems is when running graphical java applets an other java apps, that seems to cause the applets to run extremely slowly.
But I'm surprised that you are having problems with eBay due to Java though. Did you use the Mandrake Java RPM's or download Java yourself from the Sun website?