I need some clarification on (1) Is there any proof to say Mersenne primes $M_p$ are finite or infinite? if there, could you share here.. (2) If Goldbach is conjecture is true, how you can justify the statement" For all finite positive integer 2n can be written as the difference between TWO primes"
Both problems are open. For the first, see the Wikipedia article; amazingly, it is also unknown whether or not there are an infinite number of Mersenne composites. For the second, see this question and the reference to the Prime Pages; however, it is known that every even integer is the difference between a prime and a number which is either a prime or a semi-prime.