"More than half of mass shootings are domestic violence against women."
The flier states that 57% of mass shootings involve domestic violence, never says anything about women. Assaulting grandpa or a brother would fall into this statistic.
"States with universal background checks have 38% fewer women homicides."
Again flier states 38% fewer women are shot to death by intimate partners. The overall domestic violence homicide rate is not referenced, chances are if someone is willing to shoot someone they are living with they would do harm using other means.
That being said Domestic violence is a very serious issue, however it would be much more effective having more support groups for women and advertisement/education on these groups.
As far as background checks go, you need to answer a list of questions and then they run your information through a database, chances are if someone failed a background check they were unaware of the situation or lied when filling out the form which is a crime in and of itself. If they know that it is illegal for them to obtain a firearm they would just go to some hood rat and get it illegally anyways.
If firearms are something you feel need to be regulated then safety classes and the importance of quality safes would be much more effective than background checks. the majority of violent crimes committed with firearms involve stolen guns.
Just to toss in a fun little fact. US federal law states that as long as it is for personal use, anyone able to legally own a firearm can make one without any registration or serial number. kits are sold online to complete these unregistered, untraceable firearms. between being able to make your own legal firearms and criminals stealing then selling stolen guns. background checks will only stop people who are not willing to break the law in the first place.
universal background checks would mostly effect families who try to pass down guns once a family member passes or parents pass down guns once children are of legal age to own firearms themselves.