What affects "Good Moral Character" when applying for United States of America Naturalization?

U.S. Citizen wants to naturalize parent. Here are a few things that I can recall (one parent I am applying for had these situations happen): parental argument which ended in one parent being held in jail for a few hours(2002 in NJ), 2 occasions when this parent was placed in a mental institution for a few weeks and released both times(last time was 2011 in FL). (I don't know how long: used a fake ID no longer used, 2005. As far as I know my one parent's family entered the country illegally but all aunts and uncles now have papers except my parent and an older aunt/uncle. First my grandparents got their papers then they sponsored my aunts/uncles but my parent and older siblings were excluded because at that point they were over 18 years (my parent was 19). Parent has been in U.S. 20+ years. Questions: will the fact that my parent entered illegally with the family affect how they are eligible to be naturalized & how will the three occasions of situations I explained above affect their screening as "good moral conduct"; are those situations enough to bar my parent completely; essentially what are my options based on these circumstances: as I am a citizen but my parent is not?