For instance take Methuselah 140283, how can these stars exist?
I don't know but I'm pretty sure it was explained in a thread here not terribly long ago. Have you tried a forum search or checked out the links at the bottom of this page?
The error bars on the ages of these stars are quite large. Also, the ages are estimated by looking at the relative abundances of various elements in the stars' atmospheres. Small mistakes in modeling the interiors of the stars could easily throw the calculations off.
This was a problem before inflationary cosmology, when estimates of universe age could differ with a factor 2. Now the uncertainty is down to the order of parts of percent, same precision with which we date Earth [!], and AFAIK there are no longer any stars that stress the theory. Meaning the 1 sigma results on star ages cover the 1 sigma result of universe age (IIRC).
As usual, these things are just a google away: "The new Hubble age estimates reduce the range of measurement uncertainty, so that the star's age overlaps with the universe's age — as independently determined by the rate of expansion of space, an analysis of the microwave background from the big bang, and measurements of radioactive decay." [ http://www.nasa.gov/mission_pages/hubble/science/hd140283.html ]
That 2013 article covers the history of previous estimates of the age of HD 140283 (if not the history of the precision increase in cosmology).
If any one star would stress the cosmology, I think people would now start to ask for extraordinary evidence, since it would constitute an extraordinary claim.
Said another way, the physics of the CMB, from which we have our most accurate estimates of the age of the universe, are really quite simple and relatively easy to understand. The physics of stars and galaxies, by comparison, are incredibly complex and difficult to model. So when a discrepancy pops up between a CMB measurement and a star or galaxy measurement, bet on the star or galaxy measure being wrong.
It is a legitimate question that obviously confronts modern cosmology. Error bars are widely viewed as the likely suspect. Our understanding of stellar evolution is considerably less sophisticated than cosmology. Interestingly enough, a paper addressing this issue appeared today - http://arxiv.org/abs/1410.7014, Stars older than the universe and possible mechanism of their creation. It's a bit speculative, but, well motivated.
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