Should they be doing this? Imagine if HIV/Cancer/Aids/Any terminal illness was all cured and we get to a point where only old age is incurable?
Old age might be curable too. Our bodies are self-repairing and able to make replacement parts, so it's theoretically possible to either improve that ability or to augment it with technology (or both) to such an extent that people age far more slowly and possibly even not age at all.
How long would it be before the world becomes ever more populated and the global resources cannot cope with the influx of such a population increase?
I think it wouldn't make all that much of a difference if people did still die of old age. Disease isn't placing a limit on human population. It's just slowing the increase and not by all that much as far as I know. Whether or not disease existed, humans would still face global overpopulation sooner or later and have to deal with it in some way. Prosperity seems to work best - wealthy countries have smaller average family sizes and without mass immigration have populations that are stable or even shrinking.
By curing this kind of illness we are effectively giving two fingers to human nature and natural selection??
Hominids have been giving two fingers to natural selection since before humans even existed. As soon as the first hominid made the first tool, they were giving two fingers to natural selection because tools provide a way to survive and breed more successfully than natural conditions.
All that's changed since is how powerful our tools are, i.e. how forcefully we can back up our two-fingered salute to natural selection. We fart in its general direction
Alternatively, you could say that tool use and intelligence are natural to humans and that therefore anything that comes as a result is itself natural and therefore part of natural selection.
Human nature is a different question...but what is human nature? People usually use the phrase to mean "whatever I approve of", as a false appeal to authority. Maybe it's human nature to use intelligence and an aptitude for tool use to forge our own future on our own terms. Maybe it's human nature to change human nature. Maybe there's no such thing as human nature.
Obviously at a personal level it is gut wrenching knowing a family member/friend has cancer etc, and I've had a family member in the past who had cancer so can relate, I'm not completely heartless!