I've been searching around the internet for a while now, but I simply cannot find anything on how effective the Greek Linothorax was, or how effective it would be with a bronze or iron chestplate over it, what sort of protection would it have against bows? If anyone has experience on this kind of thing, could you please elaborate?
There is some uncertainty, since we don't know exactly how they were made. However, they were used, and were common, so they were probably effective.
Arrow resistance test results are reported in Aldrete, G. S. & Bartell, S. & Aldrete, A..Reconstructing Ancient Linen Body Armor: Unraveling the Linothorax Mystery. Baltimore: The Johns Hopkins University Press, 2013. In summary, they work. Quilted (multiple layers of linen sewn together) and glued construction provide similar protection to each other. 20 layers of linen, provided good resistance to arrows from 45lb bows at a range of 7.5m. Arrows from a 60lb bow penetrated effectively. To more reliably stop arrows, 25-30 layers would be better.
Other weapons were tested. The armour was more protective against them than against arrows.
According to these tests, a 20 layer linothorax was about as protective as a 1.8mm bronze cuirass. Note that the bronze test plates were annealed rather than work-hardened.
20 layers + a metal cuirass would have provided very effective protection.
The results above are generally as one might expect from tests of replica Medieval gambesons and plate armour against arrows.
Some online info about the tests is at https://www.uwgb.edu/aldreteg/Linothorax.html