Have any studies been done on the inheritability of different mental abilities such as short-term-memory, long term memory and so on.

One of the most replicated findings of behavioral genetics is that all psychological traits show significant and substantial genetic influence, however not 100% genetic influence (Plomin et al, 2016). So yes, short term memory has a heritability of larger than zero, but smaller than one.

Many studies, if at all, seem to rely on an over-all IQ. This seems extremely wide-ranging when mental abilities can be made of so many different parts. Someone with a good memory may not have good reasoning skills etc.

It is certainly true that it is possible to have good memory and low reasoning ability. However, abilities are positively correlated. Therefore, high ability X makes it statistically more likely that the individual is higher in ability Y as well, for practically any cognitive abilities X and Y (Certainly for memory and reasoning). This was discovered more than a century ago (Spearman, 1904). Although the increased likelihood may only be modest.

For example if a man has a short term memory able to memorise 10 numbers at a time and a woman has a short-term-memory able to memorise 5 numbers at a time. How many numbers could their children memorise?

There are multiple reasons why you couldn't predict in any exact manner the short term memory abilities of the child.

First, while heritability is larger than zero, it is also smaller than one. In other words, not only genetic differences are responsible for differences in short term memory.

Second, the genetics are not even determined in a unique way from the parents. Said in another way, you cannot predict the exact genetic makeup of a child from its parents' DNA. It depends on how the recombination of the mother's and father's genes. The most obvious demonstration of this is that biological siblings do not have identical DNA. Therefore, even if short term memory was 100% heritable (It isn't), you still wouldn't be able to exactly predict the short term memory of the child just from knowing the parents.

The most you could say is statistical. Parents with better short term memory will, on average, have children with better short term memory.

I seem to remember a documentary about the extraordinary memories of `native' Australians compared to their European descended compatriots. So this seems one piece of evidence that memory ability can be inherited.

Now you're getting into a socially sensitive topic of population differences in cognitive abilities. I don't know what documentary you've watched. Personally, I haven't seen the evidence that native Australians have extraordinary memory abilities. Of course, I would be open to see such evidence. However, one thing to note is that even if native Australians showed high ability in memory, that fact in itself does not prove that the difference is due to genetic differences. Memory is only partially heritable. It could simply be that due to whatever environmental or social cause that they showed enhanced memory. Of course, I'm not saying that it is impossible either.

Have any studies been done on the inheritability of different mental abilities such as short-term-memory, long term memory and so on.

One of the most replicated findings of behavioral genetics is that all psychological traits show significant and substantial genetic influence, however not 100% genetic influence (Plomin et al, 2016). So yes, short term memory has a heritability of larger than zero, but smaller than one.

Many studies, if at all, seem to rely on an over-all IQ. This seems extremely wide-ranging when mental abilities can be made of so many different parts. Someone with a good memory may not have good reasoning skills etc.

It is certainly true that it is possible to have good memory and low reasoning ability (The most extreme being some savants, e.g. Kim Peek). However, abilities are positively correlated. Therefore, high ability X makes it statistically more likely that the individual is higher in ability Y as well, for practically any cognitive abilities X and Y (Certainly for memory and reasoning). This was discovered more than a century ago (Spearman, 1904). Although the increased likelihood may only be modest.

For example if a man has a short term memory able to memorise 10 numbers at a time and a woman has a short-term-memory able to memorise 5 numbers at a time. How many numbers could their children memorise?

There are multiple reasons why you couldn't predict in any exact manner the short term memory abilities of the child.

First, while heritability is larger than zero, it is also smaller than one. In other words, not only genetic differences are responsible for differences in short term memory.

Second, the genetics are not even determined in a unique way from the parents. Said in another way, you cannot predict the exact genetic makeup of a child from its parents' DNA. It depends on how the recombination of the mother's and father's genes. The most obvious demonstration of this is that biological siblings do not have identical DNA. Therefore, even if short term memory was 100% heritable (It isn't), you still wouldn't be able to exactly predict the short term memory of the child just from knowing the parents.

The most you could say is statistical. Parents with better short term memory will, on average, have children with better short term memory.

I seem to remember a documentary about the extraordinary memories of `native' Australians compared to their European descended compatriots. So this seems one piece of evidence that memory ability can be inherited.

Now you're getting into a socially sensitive topic of population differences in cognitive abilities. I don't know what documentary you've watched. Personally, I haven't seen the evidence that native Australians have extraordinary memory abilities. Of course, I would be open to see such evidence. However, one thing to note is that even if native Australians showed high ability in memory, that fact in itself does not prove that the difference is due to genetic differences. Memory is only partially heritable. It could simply be that due to whatever environmental or social cause that they showed enhanced memory. Of course, I'm not saying that it is impossible either.