Actually they are a lot less likely to be prime than any random odd number, only 48 are now known and all have been tested up to above 13 million digits.
But yes they are very easy to prove prime compared to "random" numbers of no special form, I *think* it was proven mathematically that there is no faster proof possible, but don't hold me to that. The highest "random" number proven to be prime is only 26,642 digits vs 17 million for this new mersenne prime. There are numbers of other special forms that are also "easy" but not as easy to prove, highest one proven is 3.9 million digits.
They don't look for primes 2^n+1 because they are always divisible by 3 so they are never prime. They are looking for primes (2^n+1)/3 however, they are called Wagstaff primes, but again they are harder to prove than mersenne numbers.