A green card works the same way as naturalization or birthright citizenship, as far as leave to remain goes. The only difficulty is with a visa that is effectively a trailing spouse visa. This woman will have the choice of leaving her child behind if there is a country of origin she wishes to return to immediately, or staying until the child is emancipated at 18. At 18, it will not be easy to leave. If she has a green card and anticipates leaving at some point, she should consider applying for citizenship so that she can return for extended periods to see grandchildren, etc.
Even as a citizen or resident alien, she probably won't be able to leave the state to live, let alone the country, and horror story though her husband may be, unless he has been convicted of a violent crime related to her or her child and is incarcerated, the child's father will have visitation rights. This means she will need to stay within reasonable driving distance.
Many states are no fault divorce states, so the cause of the relationship breakdown is irrelevant to the settlement. The divorce process will be about marital debts and assets. Debts like a mortgage will have to be paid off (meaning the family home sold and equity divided). If she has been a sahm for any length of time, she may be entitled to more than 50% of the equity kr marital assets. She may be entitled to spousal support (alimony) and unless 50-50 custody is awarded, she will receive child support. Other debts will also need to be settled - credit card debt is an example, and personal loans. If her H has a personal cc that she has never had access to, she can dispute any onus to pay that off jointly.
She can expect her H to fight to get 50-50 custody and to make her life a misery via co tact about the child regardless of the custody and visitation arrangement. This is where the real fight takes place with no fault divorce, and a determined ex can use the family court to file spurious allegations for years, with allegations of parental alienation being taken seriously by the courts even though research has shown that this is pure misogyny. Free legal aid rarely extends to post divorce matters so she may well find herself on the back foot if her H retains good earning power (or if he is an attorney as my abusive exH is).
While she is making her plans, she needs to get cash back at the supermarket if she is able, and to open her own bank account asap. She can opt for paperless statements. She should also see if she can buy store gift cards for places like Walmart, so she can use them in emergencies. Walmart sells groceries, clothes, hardware, car accessories, does car services like tires and batteries, and sells necessities like TP, and has a pharmacy. If she can get herself a Target card to use for Target purchases and pay the minimum off monthly that would be a good way for her to build up credit. Getting a secret burner phone in Walmart would be a good idea if she can keep it completely hidden.
She will need to investigate whether she will qualify for food stamps and medicaid for herself and her child. There are also cash benefits available. All welfare is strictly means tested in the US. Child support paid to a parent does not qualify as earned income for the purposes of means testing iirc. The state I am in does not look at your bank balance so her share of marital assets would not be counted if her state is the same.
If she has online access to her local library (assuming there is one) it may offer access to a great many resources to improve her employability (online Microsoft suite tutorials, for instance) and some libraries have computers available for the public to use.
There may be offerings in her local community college/ junior college that would train her for a job - healthcare courses such as phlebotomy, etc, are usually short and doable. She might qualify for financial aid to do a community college course. She would need to contact the college directly to find out how to do this.
Law schools often have free legal aid clinics, and metropolitan areas often have free legal aid organisations. She should make contact asap with free legal aid for a preliminary conversation about her options and what she can expect. A free legal aid org might also be able to signpost her to aid/ support organisations.
In general, women's aid type orgs are great at moral support, most can signpost to legal aid, but there is a chasm between the understanding of abuse that she will find there and the validation she will receive, and the experience she will have in the family courts as she tries to do her best for her child. Even if she has made DV calls to the police and can show evidence of her H's abuse, family courts start with the assumption that if a father falls short of murdering his child, he is a fit parent, sadly.