Stephan,
I too had this issue a while back but my MacBook (2007) would do the same thing.
From my time on the phone with the lovely folks at Apple, I have learned that it may just be the simple fact that the software isn't keeping up anymore. I was recommended to update to more recent software but as we both know at this stage I can't go any further. I would love to be able to sync my iPhone 5c w/ my MacBook but it's just the OS X isn't current enough for that. If you have already updated the RAM to 4gb or more at this stage it appears that it may indeed be time to completely upgrade to a newer iMac. Keep in mind though, with the new hardware you have installed you should still be able to sell and upgrade to lighten the load on your wallet. The only way I was able to get mine to stop doing this was to upgrade the RAM and start w/ a fresh OS install. Still holding up pretty well but the fan does come on as if it's feeling labored but I use it for music so I built a back up fan platform to keep it cool while running for long periods of time. But other than that I don't see there being much else to do at the end of the day. I hope for you the best.