Yes, 3 and 17 both are prime factors of 51 because 51 is divisible by 3 and 17.
51/3 = 17 and 51/17 = 3.
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Prime numbers have two factors. Prime squares have three factors. Square numbers have an odd number of factors but that number varies.
No, no prime number has proper factors.
Prime numbers have two factors. 2, 3 and 5 are prime numbers.
Squares of prime numbers are the only numbers with three factors, since there must be only one nontrivial divisor for a number to have three factors. That number must thus be prime.
That's an infinite list. Numbers with three factors are the squares of prime numbers.