characterization of DC component
Statement 1, as I read the garbled syntax, is correct, assuming "without a" and "doesn't have" imply a "zero" (an answer) and not a "null" (no answer).  

The zero bin, also known as the DC bin, is the straight up sum of the signal over the sample frame in an unnormalized DFT calculation.  If the sum is zero, the average is zero.  So statement 2 is correct.

The explanation comes straight from the unnormalized definition of the DFT:

$$ X[k] = \sum_{n=0}^{N-1} { x[n] e^{ -i2 \pi { k \over N } n } } $$

Substitute in k = 0:

$$ X[0] = \sum_{n=0}^{N-1} { x[n] } $$

What is the source of these two statements?  Maybe there is a larger context to take into consideration.

Hope this helps,

Ced