How are these stars such accurate markers of the age of the Universe, independentof using expansion?

Factual Questions
1

See subject.

2

Because the Milky Way itself is not subject to the universal expansion as gravity is still holding it together.

Obviously if they can determine these stars to be 13 billion years old the universe is older (Don’t laugh; it wasn’t too long ago that the estimates for the oldest stars exceeded the estimate of the Universe’s age.) And since they have theories about how long it took the first stars to form, they can add the appropriate amount. I presume they can tell that these are Population II stars by their spectra (low in elements above Helium) so they know they were formed early in the life of the universe. (There are even earlier Population III (yeah they’re numbered backwards), but I don’t believe nay of those have been observed rather than inferred.)

3

A white dwarf is the remains of a star which has used up all available fuel, and has stopped producing energy from nuclear reactions. As such, they very slowly cool down, as they radiate energy. The dating method is to determine it’s temperature. There is a theoretical model which tells us how long it should have taken for the white dwarf to reach the measured temperature. The coolest ones measured are at about 4,000K, giving an approximate age of 13 billion years.

This check is completely independent of the expansion of the universe.

4

Thanks. Any reason these stars in the image are the best found for this type of analytical determination?

The cite makes it seem so.