The boot failures sound like a hardware problem. The fact that it couldn't cooperate with certain memory tends to support that conclusion too, although memory issues could be coincidental.
I'd recommend running Memtest86 on it. Memtest86 runs as a bootable disk that takes over the computer, so there will be some downtime but well worth it if it finds a problem. The full set of tests takes hours, but my experience has been that a truly flaky data path will be detected before it finishes the first pass of the first test. My current practice is to throw Memtest86 on a new machine before I bother doing anything with it, and I routinely come across flaky mobos/RAM. At work recently it found slightly flaky memory in a pair of 1 GB machines before they went into production as an electromagnetics simulator; finding that one in production would have been hellish.
If you want to keep the machine live, you can try building large projects. E.g., Linux kernel. The "-j" option of make lets it run multiple processes at once which can drastically increase system load, especially on multi-processor machines. OTOH, finding a problem on a live system stands a good chance of blowing away the filesystem. Memtest86 or scratch drives might end up taking less time than blowing away the production partitions.
If you do find flakiness, the first thing to do is turn off the machine (using a power strip to avoid the @#@#$!$#$# always-on power supply), and systematically unplug and reconnect every electrical connector. CPU, RAM, fans, backplane/daughtercard, SCSI card, power connectors, you name it. Reseat every single connector. I've seen this fix a lot of "funnies". In fact, you might try doing this preemptively as the downtime would be fairly small.
I don't want the world, I just want your half.
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