Is a felony for attempted robbery relevant in a custody battle if the felony is over 14 years old?

I'm 30 years old, and I was convicted of attempted robbery when I was just 16 (but no y.o. status). I have a few other arrests on my record but nothing that ever stuck, besides that felony. Is it relevant in a custody battle? Also, in NJ, where I was living with my son and ex girlfriend, there was an incident where she lied to have me arrested for domestic violence but everything was dismissed. NJ DYFS (child services) got involved since a child was present at the time of her allegation. They've been monitoring the situation to this day and have given her and I drug screenings to where she has tested positive for marijuana, and I have tested negative twice. She has also been uncooperative with her DYFS case worker. Through all of this, she has since moved 2 strangers (men) into her home where she lives with my son as well, which I consider to be a form a negligence and endangering the welfare of my child, aside from her drug abuse. I now have my son in my possession and will file for full custody. I have proof from the landlord of her moving 2 strange men into the home. She has been on welfare for about 4 years and collects cash, food stamps and Medicare for my son, and her other son who I am not the father of and who also has been living with my ex's mom for over 2 years. Welfare has put me on child support when my son was born, but they had no knowledge of the fact I was living with her and my son the entire time taking care of them while paying CS at the same time. They also paid the rent for over a year. So, 1) will this result in a welfare fraud case against her? 2) will the court consider it a form of child neglect or endangering the welfare of a child for moving 2 male strangers into the home around the child? And 3) will the felony conviction against me over 14 years ago be relevant in the child custody battle?