Say you are an alien, and you want to know roughly the male-to-female ratio of people. Let's say the true ratio is 50%.
Wouldn't this be done by an unbiased sample that's quite small, regardless of whether there's 100M or 8B people on the planet?
In not extreme cases when your population is much bigger than the sample size you're correct that it doesn't really make any difference.
When I was an undergrad first learning statistics I asked my stats instructor (a grad student) about this issue and they responded with something like "the population size doesn't matter because for the assumptions of the test to be met... such and such..." I kind of accepted that answer — we were talking about asymptotic inferences — but it never seemed quite right to me.
The example I gave was actually motivated in part by a sort of real-world problem I was dealing with: let's say you only want to make inferences about a population of 20 individuals. Certainly if you have a sample of 19, the confidence about the population will be much stronger than if your population is 100 million.
One thing he did say which is probably right, is that that 1/20 you didn't sample might throw things off, so it's more influential in a sense than a single member of a population of 100 million.
At the time I hadn't learned about exact and Jaynesian-permutation statistics, but that's probably the right way to think about finite populations. That is, something like "what are all the outcomes you could observe, and what proportion of those does my observed result represent?"
It's just that usually our population is so large that the exact test approach becomes infeasible to deal with without approximations, and you end up with the typical classical asymptotic statistics.
It's all maybe a moot point but it's always a good idea to think about the population you're trying to make inferences about. I think that probably includes the population size, and I think population size is probably bigger than you might initially think sometimes.
As for your last question, obtaining an unbiased sample is kind of harder as the number of attributes you're being unbiased with regard to increases. It's a permutation problem again, probably implicit usually with regard to sampling representativeness.