logical fallacy where the criticism can be applicable on both sides?

The two examples are cases of double standard .

The comic case is a case of false dilemma . The argument implied is that "EITHER you live outside of society and criticize it OR you live as part of society and you shut up", which of course is bs.

These are not logical fallacies. At best they are examples of manipulation or psychological projection and blatant hypocrisy.

There happens to be an excellent book on the subject I highly recommend. It's called Logically Fallacious written by Dr. Bo  Bennett. It's a must-read. 

They are not fallacies because, in example 1, the reason provided does make sense. Information is being withheld, so perhaps this is an example of evaluating an argument without being presented or considering all the information (like who else has tomorrow off).

In example 2, perhaps X is being hypocritical, or they cannot volunteer for some good reason, or they are also including themselves in the "more people." The fact that Y doesn't volunteer doesn't seem at all relevant, since Y is not suggesting people should volunteer.

I would recommend spotting implicit statements using the principle of charity. In example 1, there seems to be an implicit statement: whoever has the day off tomorrow has to do the dishes today. If x claims that he/she does not need to do the dishes either day, then that would be a violation of the rule that both of them might had agreed on. 

It sounds like x either adjusted the schedule to do the dishes on the same day as y, or there was an honest mistake. If x did that by mistake, and if he or she follows through with the rule, then x will do the dishes tomorrow. Not doing that would be special pleading.

I think there's another implicit statement in example 2 and a different fallacy being committed by y. But keep in mind that implicit statements are added by us. This is why we should try to represent arguments in the best way that we can.  

I suppose one could stretch this into a claim of tu quoque, but other than that, I see only banter.

[Tu quoque (“you too”) is an appeal to hypocrisy. The argument states that a specific position is false or wrong and/or should be disregarded because its proponent fails to act consistently following that position.]