The general difference between "taking" and "taking on" responsibilities in these three cases is:
when you take responsibility for something, it generally means that you accept responsibility and act in an appropriate way by, for example, doing your job (as a parent or a student or a teacher or a banker) as well as you can and don't rely on others to do your work for you.
when you take on responsibility for something, it generally means that you assume new responsibility for something you aren't or weren't expected to be responsible for.
There are other ways to use these terms, but all these sentences are about personal and familial responsibilities, not about social or political responsibilities.
"DINK couples refuse to play the role of parents and (take / take on) the responsibilities to cultivate offspring."
DINK couple: "a couple with two incomes and no children". They don't have children, so I'd use take on the responsibilities (because they don't want to have them).
"Mother is getting old, but I haven't (taken/taken on) the responsibility for supporting her."
This is complicated because it's a Chinese cultural problem. If you're the eldest son, then your mother is your responsibility and you must take it (accept it) or violate cultural norms. If you're not the eldest, then you're talking about not assuming (taking on) someone else's responsibilities for your mother. In my wife's family, all six daughters (no sons) contribute equally to the support of their parents, so each has taken on a certain amount of responsibility even though she doesn't necessarily have to, and the eldest daughter cannot afford to support them on her own.
"She is ready to (take/take on) the responsibility of a wife bravely. (Note: Her husband is seriously ill in the hospital.)"
This depends on whether she's a new wife or a veteran wife of this husband. If the marriage is new and the illness is new, then I'd use take on because those kinds of responsibilities are unexpected for new marriages. If, as in my case when my late wife needed someone to be her daily caregiver during the last nine months of her life (she had cancer), after eleven years of marriage, I would use take the responsibility of a dutiful and loving husband. It's part of the marriage vows, but even though almost no one is prepared for such a thing, it seemed to me the only thing to do even though I could have afforded to pay someone else to do it. Had I paid someone else, I would have been shirking my responsibilities, but by accepting them, I wasn't adding anything unexpected to my responsibilities.