I don't understand #1. As I understand it ISO controls the sensitivity of the sensor so without also selecting to overexpose the image it won't be overexposed just by increasing the ISO. So the question still remains of what exactly increasing the ISO does. The camera doesn't limit you to lower ISO but to higher ISO.
As it stands I think there's some tinkering with the actual electronics involved here that can't be achieved in software alone after the image is taken. What I'm trying to figure out is what the ISO does as I'm not aware of any effect besides increasing noise. So I don't know if shooting at a higher ISO will have an effect when adjusting in software. As I stated the problem isn't just overexposure in these areas but also an increase in noise which makes it look really awful.
At the hardware level, the signal coming from the photosite is an analogue one. Before this signal (voltage corresponding to number of photoelectrons captured by the photosite) is converted to a digital number by the ADC, the signal is amplified. The purpose of the amplification is to increase the signal level to reduce the relative magnitude of the quantization error, and possibly read noise (depending on the sensor design). The amplification of the analogue signal is called the
gain, and typically (traditionally?) this gain factor is what is increased when selecting a higher ISO setting.
Things got a little more complicated after Sony started producing sensors where the ADC is integrated into the sensor itself, which resulted in the so-called ISO-less cameras, such as the Nikon D7000 (and many others, since). With these sensors there is little benefit to adjusting the analogue gain on the sensor --- you can just as well simulate a higher ISO by multiplying the digital number (after the ADC process) by a factor > 1.0. Strictly speaking, these sensors are only approximately ISO-less. On these sensors, the method I described in my first reply (above) makes perfect sense, since you effectively lose nothing by using a lower ISO (lower ISO followed by boosting in post, vs. higher ISO, in terms of visible noise). On the other hand, using a lower ISO effectively prevents clipping of highlights.
To get back to your original question: I have no idea what SX50's DR correction does, but it is most likely what Nikon calls "Active D-Lighting" on its DSLRs, and what Canon calls "Highlight Tone Priority" on its DSLRs. Although it seems strange that this option restricts you to a higher ISO, it is functionally consistent, i.e., if you are shooting at ISO 200 with DR correction enabled, the camera would choose to keep the shutter speed fixed, and drop the ISO to 100. If, say, ISO 100 is the lowest ISO setting of the sensor, then the camera
must limit you to ISO settings of 200 or higher when you turn on DR correction, in order to give itself room to "drop the ISO".
Let us assume you set the camera to ISO 200 with a shutter speed of 1/100 s, with DR correction enabled. You could produce similar results by keeping the ISO fixed at 200, and doubling the shutter speed. This would produce exactly the same amount of underexposure (ISO 200 at 1/200 s produces an equivalent exposure to ISO 100 at 1/100 s), but doubling the shutter speed affects other aspects, such as the degree to which motion is frozen, hence the camera fixes the shutter speed, and drops the ISO "behind your back".
I would recommend reading up on the sources of noise for further insight. Understanding photon shot noise will make the operation of "DR correction" clear.