The dynamic strength/toughness causes no special situation. The X represents a toughness number just like a 5 or a 10 would, the only difference is that the value changes based on the circumstances. However the X still represents a value regardless of the circumstance. Therefore gates affects AWSN the same as it affects any other character.
Interesting, my understanding was that cards that have dynamic numbers aren't really able to have changes to their numbers because it would dynamically reset to its proper stats. Maybe can we get an elder confirmation on this?
You can indeed adjust the numbers on characters with dynamic abilities. If a character has static ability (like all normal cards), you do not say that it cannot be adjusted because the number is printed on the card. Rather, you take the number printed, add the positive changes and subtract the negative changes to obtain the final ability value.
Similarly, in Type 2 a character with dynamic abilities is still considered X+1/X+1 once the character succeeds in battle and obtains an experience credit. You don't disregard that credit, it is added to whatever X happens to be at the moment.
So essentially, any X abilities on characters are added to all increases and decreases, the same as for any other character. If the result is 0 or less for toughness, the character is immediately discarded.