In the last two years I guess I have seen more of this marginal RAM where the most common symptom would be reporting half the capacity of a DIMM, sometimes a pair.
But yes RAM issues can have profound side effects. And the older Fully-Buffered had to have those heat sinks because they can run too hot without. Apple even advises to wait 15 minutes before opening the side door, and that just the sudden change in temperature can be a shock, and harmful.
Heat can stress and age RAM and we have never had or seen RAM this hot. I replaced my Apple RAM when it was about 2 yrs old.
Even PCs have their own problems and DDR3 can be fussy, or so it seems. Mostly, I try to stay with Crucial as much as possible, but even there they have had their own issues, sometimes a change in batch or construction design.
A chip plant having to shut down or hit by storm can take weeks or longer to get back online and back to reliable batches, too. Which happened in fall of 2000 I think.
Mixing different batches, even from the same vendor or factory can even have problems and not get along. You could have two or three sets that are fine until used together. Which is why the ideal is to buy and replace in full sets, or take a chance.
But if Riser, Apple has been known to advance exchange if taking it in isn't easy. And easier to take or send in a Riser than the MP!
Riser for 1st gen
http://www.dvwarehouse.com/Apple-Memory-Riser-card-for-Mac-Pro-First-Generation- 922-7695-----NEW-p-37247.html
PSU
http://www.dvwarehouse.com/Mac-Pro-Power-Supply-c-247_4884.html