Is it a true statement to say that: If two statements are NOT equivalent they will NOT imply one another?
Doesn't "imply" mean they are equivalent?

It is true that if they both imply each other they are equivalent.

However, one statement could imply the other without them being equivalent. Consider the statement: "If you are a student in Math 2283, then you are a student at University of Minnesota." That is a true statement. However, it isn't true that "If you are a student at U of M, then you are a student in Math 2283." So the smaller statements "you are a student in Math 2283" and "You are a student at University of Minnesota" are not equivalent, but one implies the other.

In writing the negation of "If all pigs can fly, and some birds can not fly" Should I break the phrase into 2 parts (If all pigs can fly) & (some birds can not fly), or into four parts (all pigs) (can fly) & (some birds) (can not fly)?

Well, "all pigs can fly" is a single statement in and of itself. Since we couldn't assign a truth value to the two words "all pigs", we can't break it into a seperate statement.

If you are converting the phrase "all pigs can fly" into mathematical symbols, you don't necessarily need multiple symbols. On the other hand, if you would like to make sure you are negating "all pigs can fly" correctly, having several symbols (such as a for all symbol along with a statement) would be helpful.

What are rules for negation with the quantifiers?

To negate quantifiers, think about how you would disprove a statement. If I said "For all n in the Natural Numbers, P(n) is true" and you wanted to disprove it, all you would have to do is find a single n where P(n) is not true. So for all n P(n) switches to there exists n such that ~P(n). Likewise, you flip there exists to for all. In the course book Note 2.3 on page 8 talks about negation of the quantifiers.

Does the "either" in problem one refer to the comparing i) and ii), or the individual phrases in i) and ii). i.e. comparing: [(P or Q) ==> (P and R)] with [(P or ~R) ==> (Q and ~P)]. How can you tell if one statement implies the other by looking at their truth tables? If they match in all cases, does that mean they imply one another?

"Either" refers to comparing i) and ii). One technique for seeing if any statement is a true statement, is to add a column for that in the truth table and see if under all possible conditions, that statement is true. So here, we could make a column i) ==> ii) and look for all trues.

If two statements match everywhere, then they are equivalent statements. (Note: equivalence P <==> Q is the same as (P ==> Q and Q ==> P) so if P implies Q and Q also implies P then they are equivalent statements.)