How can a very old star be formed among Population III stars? After looking at some articles about old stars, (this, that, and this Wikipedia article), I thought about how such an old and small star could have formed. As the early universe had very massive, early-type stars and (no? almost none?) smaller, late-type stars. So how could a ~$1M_\odot$ star have formed in the early universe, where massive Population III stars dominated?
Here are my theories, if you may find them helpful.
Mass loss via a long-dead companion
Star Foo and Star Bar are both $300 M_\odot$ stars closely orbiting each other. Star Bar is slightly more massive than Star Foo, causing Star Foo to lose mass to Star Bar. Now, Star Bar is $580 M_\odot$ and Star Foo is now drained to $20 M_\odot$. As Star Bar ages and becomes a supergiant, Star Foo gets engulfed by Star Bar, reducing Foo's mass to $\text{1-2} M_\odot$. Star Bar then goes supernova, ejecting Star Foo. 13 billion years later, Star Foo ages and becomes a subgiant, what we see today.
Formed in a massive star's disk
When the universe was young, there were plenty of materials to create stars. One massive star, star Foo, formed. It had a vast disk of gaseous material, weighing over $2 M_\odot$. In the disk, perturbations caused a massive clump of gas to form together, causing the ignition of hydrogen and creating a small, yellow dwarf of mass $0.8 M_\odot$, called star Bar. When Foo went supernova, Bar was ejected. 13 billion years later, we see an ultra-metal poor, very old, late-G type borderline-subgiant.
Split off from a massive star's envelope
A massive star forming region creates a star, Foo. Foo is very massive and large, and begins losing mass through the stellar wind. In a massive eruption, over $1 M_\odot$ is ejected from the star's envelope. This gas clump collapses under its own weight to form a small, Sun-like star. As the universe ages, we see a metal-poor, very old red giant.
Many astronomers and cosmologists conjecture that Population III stars were very massive, perhaps even extremely massive. While this opinion is not universal, it does appear to be dominant. If this is the case, at least some of the very first stars died shortly after they formed, within a million years or less. This in turn would have meant the seeds for the first Population II stars would have been present in some places shortly after the very first stars formed and then died.
The Population III stars did not all form (and die) at the very same time, and those seeds needed to create the first Population II stars would not have been dispersed uniformly. So at least for some period of time in the early history of star formation, there would have been a mix of Population III stars in some places, Population II stars in others.