In response to question (1), an authoritative source is Peter Rudman in "How Mathematics Happened: The First 50,000 Years". Some revelant quotes:
On the Ishango bone (20,000 BCE):
The concept of division, which must precede the concept of prime number, probably did not evolve until after 10,000 BCE and the emergence of herder-farmer cultures. The concept of prime numbers was probably only really understood after about 500 BCE by Greek mathematicians.
On the Babylonian clay tablet Plimpton 322 (1800 BCE):
This clay table shows that Babylonian scribes understood Pythagorean triples and perhaps the Pythagorean theorem. It also hints at some understandig of number concepts: prime numbers, composite numbers, regular numbers, rational numbers, and reduced fractions.
On the Sieve of Eratosthenes (250 BCE):
Is easy to apply and to understand. Babylonian scribes could have invented it more than one thousand years earlier --- but they apparently did not. Its invention was only possible after Pythagoras (500 BCE) and Euclid (300 BCE) had made the study of properties of numbers a subject worthy of the attention of Greek philosophers.
In response to question number 2, as described by O'Connor & Robertson, see also the Wikipedia entry, Islamic mathematicians were the heirs of the Greeks throughout the Middle Ages, motivated in part by their interest in practical applications of geometry and number theory to architecture and decoration. (Similarly, the Islamic law of inheritance served as a drive for the development of algebra.)
The translation by Islamic scholars of the mathematical works of Greek mathematicians was the principal route of transmission of these texts to the Middle Ages. For example, Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912), while the Latin translation had to wait until Xylander (1575).
Two
Some notable Islamic heroes of prime numbers:
As noted by Stopple, the 9th century astronomer Thabit ibn Qurra studied prime numbers of the form $3\cdot 2^n-1$ (now called Thabit numbers).
Ibn Al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form $2^{k-1}(2^k - 1)$ where $2^k - 1$ is prime. As noted by John Stillwell, Al-Haytham is also the first person that we know to state the theorem that if $p$ is prime then $1+(p-1)!$ is divisible by $p$ (only proven 750 years later by Lagrange).
Al-Farisi (born 1260) stated and attempted to prove the fundamental theorem of arithmetic, on the unique factorization of an integer into prime numbers.
Finally, the "why" question: There are no comparable heroes to Thabit or Al-Haytham in Mediaeval Europe. My surmise is that this is because Christianity, with its figurative art, did not stimulate the interest in geometric and numerical patterns to the same extent as Islam did.
