Nash Equilibrium and Dominant Pure Strategy when payoffs identical What happens if the payoff is the same across strategies? I have some thoughts but I am not sure.

$$
\begin{array}{c|cc}
 &amp;C&amp;D\\
\hline
A&amp; 1,1&amp; 1,1\\
B&amp; 0,1&amp;1,-1
\end{array}
$$

Dominant strategy
By definition, a player's strategy X is dominant over his other strategies if strategy X is the best strategy regardless of what the other player chooses as his/her strategy.

For player 1 (Row), is ‘A’ a dominant strategy for player 1? I'm not so sure here. Yes, A is the best response if player 1 plays C, but both A and B are the best responses to player 1 when he plays D. In this case, should strategy A be considered a dominant strategy for player 1?

Same goes for player 2, if player 1 plays A, then player 2 can respond with either C or D. If player 1 plays B, then player 2 respond with C. Can i say C is a dominant strategy for player 2 then?

Nash Equilibirum
Nash Equilibrium is defined such that no other player can do better by unilatarally change his strategy. 

Is (A,C) Nash Equilibrium? By definition its Nash Equilibirum since player 1 will not change from A to B since payoff of 0 is not better than 1. player 2 will not change from C to D, since payoff of 1 and not better than 1. (Is this thinking correct?)

Is (A,D) Nash Equilibrium? By definition its Nash Equilibrium since player 1 will not change from A to B since payoff of 1 is not better than 1. player 2 will not change from C to D, since payoff of 1 and not better than 1. (Is this thinking correct?)

Thanks.
First, make sure you have a definition of strategy clear. Strategy describes the actions a player chooses conditional on information etc. In your case it's a one-shot game so perhaps the distinction between an action and a strategy is not that important.

In game theory there is the notion of a "weakly" or "strictly" dominant (or equivalently "weakly" or "strictly" dominated) action. Nash equilibrium does not allow for any strictly dominated action to be chosen. It can be, however that a weakly dominated action is chosen. Hence you cannot cross out a whole row or column because an action is weakly dominated.

Your thinking regarding to NEs is correct. In both AC and AD no player can be strictly better off by changing his/her action conditional on opponent's action being held constant.

Finally, game theory done properly is rigorous enough (at least at undergraduate and graduate level), that there is never grey area. The definitions you gave are not stated mathematically and not 100% correct either, hence they seem a bit vague.