It looks like you have a number of probably unconnected problems here.
If the memory check is repeating over and over it points to a hardware problem - the PC hasn't even looked at the hardisk at that stage, so software is irrelevant. The check that is disabled in your BIOS is probably the long check, the PC will still do a quick check regardless. It would be worth enabling the long check to see if it throws up any problems. It could be a motherboard or CPU problem at that stage, so don't go throwing money at RAM based on that test alone. The next step is to substitute components one by one and observe the results, or to use a commercial diagnostic program on it.
Given your sterling services to the wannabe community, I'm sure you could find a grateful IT professional / ATPL wannabe who lives locally to come round and lend a hand (ooh, er, I was dangerously close to volunteering there!).
Turning to the software side of things, bear in mind that ME requires a lot more resources than 95/98. Although you have a reasonable spec PC it will run noticeably slower on ME. Unless you need the features Windows 98SE or Windows 2000 Professional would be a better bet.
Can't comment on your dual booting woes, as I am a Micro$oft disciple and am not versed in the black arts of Linux :-)
Seriously, if you need a hand give me a shout, I'm just up the road.