Nash Equilibrium and Dominant Pure Strategy when payoffs identical
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.