The original statement is equivalent to "It is not the case that there exists 0 such that for all x, x+0=x", so, to form negation, you just remove "It is not the case that".There does not exist 0 such that for all x, x+0=x

Negation: For no 0 does there exist x, where x+0 =/ x

Should be "accurate or entertaining".The newspaper article was neither accurate nor entertaining.

Negation: The newspaper article was accurate and entertaining.

You are making this more complicated than necessary. "There exists epsilon > 0 for each N" means "For each N there exists an epsilon > 0 (which may depend on N)", and it's a wrong way to start the negation. Rather, you keep the same order of variables: epslion, N, n, but change the quantifiers into the dual ones. "There exists an epsilon > 0For each epsilon > 0 there exists N in the Natural Numbers such that for all n >= N, the abs|An - L| < epsilon

Negation: There exists epsilon > 0 for each N in the Natural Numbers such that there exists n >= N, |An - L| >= epsilonsuch thatfor each N in the Natural Numbers, there exists an n >= Nsuch that|An - L| >= epsilon".

The rest seems good to me.