A) S1 = 1 - 1 + 1 - 1 + 1 ... = [0] or [1] depending on how many terms you use.
Thus, S1 for n terms is 0 if n is even or 1 if n is odd.
The average result is 1/2, but you never ever get the average result, you get either 0 or 1.
B) S2 = 1 - 2 + 3 - 4 + 5 - 6 = [n / -2] if n is even or [(n+1) / 2] if n is odd.
The average result is even stranger. If n is even, the average result is 0. If n is odd, the average result after many terms approaches 1/2. So the average of the average is 1/4. But again, you never actually get 1/4. You get [n / -2] or [(n+1) / 2].
In the video, they added S2 with an even number of terms to S2 with an odd number of terms. So, they aren't actually playing by the rules. They are using a sum of terms which is neither odd nor even. Either n is even or n is odd, but they use both.
C) From [S - S2 = 4S] they get [-S2 = 3S]. But, you can only do that math if S is not infinite. In other words think about this: let Y = X + 1 and Z = X + 2. If X is infinite, then you get Y is infinite and Z is infinite. Thus Y = Z. Thus, X + 1 = X + 2. Subtract X from both sides, and you get 1 = 2. The problem is that you can't add or subtract infinity from both sides (just like you can't divide by zero on both sides and get anything meaningful). So they are fudging again if S happens to be infinite.
So lets assume that S is not infinite. In that case, S = -S2 / 3. Lets look at -S2 / 3. That value would be -S2 / 3 = [n / 6] if n is even or [(n+1) / -6] if n is odd. With an infinite number of terms, n, you get [-S2 / 3 = infinity] or [-S2 / 3 = negative infinity]. Thus S is infinite, which violates our assumption. So, you can't go from [S - S2 = 4S] to [S = -S2 / 3].
In conclusion, if you average a number S1 that can't be averaged, add in the average of an average S2 that also can't be averaged, then forget whether you are using an even number of terms or an odd number of terms, and finally subtract infinity from both sides, you will end up with 1+2+3+4+5+... = -1/12.
That or I've had too much wine.