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I only just got the replacement KT7. The green lines seem to be gone. Now the system just resets itself every 2 minutes or so. The processor temp is showing as normal.
Update: Cleaned out Windows and tried a reinstall. It rebooted near the end of the installation, before it was supposed to. Suggestions for this, or do I hire an exorcist? >>
My last mobo, which I used actively and extensively until about a week ago, was an Abit KT7 (KT-133). So I have what I hope is some valuable information.
OK--scatch that. I see you are running Win98 SE. I had some particular problems with the KT7 and Win XP relating to the infamous nVidia "infinite loop" error, and how I solved it--forever--on my KT7 (seriously, it never returned--not even once.)
But as you are running Win9x, none of that applies to you.
I ran the Abit KT7 successfully for several months under Win9x and had no serious problems that I can recall--including running the fsb and the DRAM bus asynchronously, at 100MHz and 133MHz, respectively.
A few things, though: I did flash to the latest official (I'd stay away from the beta bios's) bios (which is 64, I believe). Also, the KT7 is a terrible front side bus overclocker. This is not because of the board in particular, it's just that the KT133 chipset as opposed to the KT 133A variant that appeared later, simply did not support fsb speeds much over 100MHz very well at all (Whereas the KT133A runs normally with a synchronous fsb and DRAM bus of 133MHz +) As I recall, even when I tried to overclock the fsb as little as 5 MHz it would run OK for awhile--maybe an hour or so--and then I'd start having problems and errors. So I wound up not overclocking the FSB at all and running the DRAM bus at a straight 133MHz, and did so very successfully for a long time under Win9x.
I do recall a couple of times seeing GREEN LINES appear at the top of the screen (actually it was more like ONE green line running horizontally at the top of the screen.) Seeing this usually was precursor for a system lockup or other failure which would follow within minutes, predictably. As I recall, I narrowed it down to some BAD DIMMS I had been using in the system. It was old ram, actually, and PC-100 rated at that, which I had managed to push to 133MHz until, I suppose, the ram faulted obviously. Replacement of the DIMMs got rid of the green line syndrome for me. I remember in particular once where a particular DIMM was concerned that I noticed the absence of the green line when I took it out, and noticed it appear when I put it back in. The position of the DIMM (which slot it was in) had nothing to do with it. It was simply bad, and I got the "green line" whenever I put it in. That pretty much solved the problem of the "green line" for me.
Now, regarding your problem--as it seems you have a replacement board in hand and odds are highly against your getting a second defective board in a row--I'm going to say that your problem is in your power supply. In fact, I've read many such cases in the KT7 forums over the last year of inexplicable reboots that were cured by the installation of a better power supply. That's sure what it sounds like to me.
The problem with power supplies is this: you can't judge them by their rated output. You have to judge them by their quality, which is difficult to do based on scarcity of power-supply reviews....

I think Anandtech once did an excellent power-supply "round up" which demonstrated quite convincingly how power supplies with rated outputs of as much as 100 watts less could outperform their "more powerful" competition by a convincing margin. In fact, I spent some time today looking for that review, but could not find it, so it may not have been an Anandtech review--although I could have sworn it was. Anyway, their number #1 choice for power supplies was Antec. I'd have to say I agree. You might wish to check into this further, as your symptoms surely sound like an insufficient power supply--quality-wise. It's important to recall that many PS brands advertise outputs that tests reveal they do not actually, in some cases, come close to. So, I'd say that if you can get yourself an Antec 300-watter, which is what I used with the KT7 for a long time successfully, you can at least and if nothing else eliminate the Power-Supply part of the equation.
In summary: with decent ram, no fsb overclocking, and a decent power supply, I had few problems with the KT7 under Win9x running a 100MHz fsb and 133MHz DRAM bus. And, always running a nVidia videocard to boot.
Good luck and the best to you!