Ok, thanks for taking the time to answer those questions. What I believe we've got here is two separate but possibly correlated issues. Here's my theory:
Your answer to #1 tells me relatively definitively that there is something wrong with your PSU. What exactly that could be is difficult to say without you running a multimeter test or me having your rig in front of me. Restarting could be caused by both too much power and too little power, but I'd be willing to bet that in your case it's the former.
My reasoning here is twofold. First, the faulty BIOS readings appear with both the good and the bad PSU. That tells me that there is a problem with the sensor on the motherboard. Second, since I'm assuming that the sensor used to work correctly, I must also assume that something may have happened to it in order to make it fail. Undervolting would not do this, but I'd imagine that high levels of erratic current could. Basically, I believe that your board sensor has been fried by incorrect high power flow.
The error makes the sensor completely useless as a tool; it could read 1k volts or .0001 volts and it still wouldn't tell us what's going on. However, the actual incorrect number is of no consequence; what matters is the limitation it puts on your ability to assess the situation. I believe that this erroneous reading is masking the true voltage fluctuations occurring in your system. They may not be as high as the erroneous reading or they may even be higher (unlikely since something probably would have exploded), but we can't be sure. Regardless, you have a power problem.
Those probably high levels of power are likely tripping a failsafe on your board or CPU. I couldn't tell you exactly what the trip point is or even how it operates since I'm short on time and unable to do that level of research right now. Whether that failsafe is being tripped by the issue directly (solely because of voltage) or indirectly (due to the high temperatures created in certain areas by very high voltage) is hard for me to say and is, frankly, irrelevant. What matters is that something bad is happening and that it has something to do with your PSU.
It is important for me to note that it is possible--unlikely in my mind, but possible--that the erroneous BIOS reading and the PSU's failure are entirely unrelated and you are just a victim of unfortunate circumstances. However, whether the two issues are related to one another is secondary to the shared end result: your computer doesn't work properly with your PSU.
That, then, brings us to the question of where this leaves you. The easiest solution here is obviously to purchase a new, better-quality PSU. You can find 80+ Bronze units
like these for around 50 bucks. That's a quick, simple, easy fix and it would be the route I'd take. I know you mentioned earlier that you can't afford a new PSU and perhaps this isn't any of my business, but if you can't come up with $50 for a repair then you may have more pressing issues than your computer.
That said, there is one other thing you could try: cleaning your PSU. I believe that all roads will lead to having to swap out your PSU, but I suppose it couldn't hurt to exhaust all available options if you are so inclined. You mentioned that your computer is very dusty--dust is the arch nemesis of proper PSU operation. If the PSU is overheating or if its fan is unable to spin it could be tripping a failsafe (if it has one) and shutting off, causing your restarts. I don't think this is as likely as the PSU's power output being borked, but it's easy enough to clean a PSU and there is always a chance that I'm a moron and am completely wrong. It never hurts to double check.
Outside of that, there's not much else I can suggest. I'm not exactly an electrical engineer, so the nitty-gritty of PSU operation is not something I'm comfortable delving into. However, I
can tell you from a hobbyist's perspective that operating with a faulty PSU is extremely dangerous to your system and should be rectified as quickly as possible.
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