The ideal solution for you would be, of course to have a fully functioning 60GB PS3 last forever. This would give you both things you want: your saves and your backwards compatibility.
Reballing your PS3( re-soldering the chip joints )
Look for a reballing service. Ideally, the service will use more durable leaded solder instead of Sony's lead-free solder. It's likely to last many years.
So you're looking at the initial reballing cost (roughly 125$USD) + likely future repairs. This would meet both your goals.
Buying a PS2 and Transferring your Saves
Another option is to spend 100$USD or less on a new PS2 to play your old games.
Then sync your trophy data with the PSN server, and use a external HD or Thumbdrive to transfer all game saves and small data to your wife's PS3. Most saves are not copy-protected. You will be able to see this my inserting a formatted thumb drive into the PS3 and looking for a copy operation. I believe Skyrim saves can be copied, I am not sure about Disgaea 3.
Additionally, if you have PS+, your can sync your saves to the cloud, then pull them to your wifes PS3. Please take note that Sony holds your saves in the cloud for 24 hours before you can pull them to a new machine.
This solution would achieve both your goals and cost less, but you'll have another piece of hardware. Personally I believe this is a better long term solution.
Edit: Here is a topic attempting to list all save-protected games: http://www.ps3trophies.org/forum/general-ps3-discussion/62819-ps3ts-official-locked-saved-games-thread.html
This could as already stated be a sign of a hardware issue in the PS3 itself, However don't overlook the possibility of it being a television problem. Try hooking up to another tv, and if that does not fix the problem there may even be something up with the AV or HDMI cable you are using. if swapping out these things does not help and you are still under a warranty I would recommend downloading all your save files to flash drives and return the ps3 for a new one.
as for the heat problem try to keep your PS3 in a well ventilated area. if this does not help it is more than likely either due to some sort of heat sensor problem or something to do with the fans.
Best Answer
If you have a PC video capture card, you could route the TV output to it.