1. Think of it this way: a multiple of 6 is in the middle and you've got three other numbers on each side. Since we're talking about consecutive numbers, then the other 6 numbers apart from the median will not be divisible by 6. Numeric example:
3 4 5 6 7 8 9

2. The sum of consecutive numbers can be calculated by using the standard formula for progressions (after all, it is a progression of ratio 1).
Say your smallest number is a: then the sum will be a*(a + 6)/2. This will be divisible by 6, meaning that a*(a + 6) will also be divisible by 6. Since the difference between a +6 and a is exactly 6, then this means that a HAS to be be divisible by 6. Think of picking any other number that isnt't divisible by 6, add 6 and then check to see if the product of the two is divisible by 6:
If x = 1, then x + 6 = 7 and 1*7 is obviously not divisible by 6
If x = 2, then x + 6 = 8, and 2*8 is again not divisible by 6.

This happens because of the "ancient" rule that if a is divisible by b and c is divisible by b, then a - c will be divisible by b.
So 2 is sufficient as well: you get that the first and the last numbers are divisible by 6.
However, I don't think that this is an official GMAT question, since the two stmts yield different answers. IMHO, you should try to solve questions from trustworthy sources.