Impossible scenario.
While Section 8 is frequently derided in the fashion referenced above, it's very difficult to play the system in the manner described due to the low supply versus tremendous demand for housing and vouchers among those in need of it. Applicants typically have to spend years on waiting lists before Section 8 housing becomes available, and in many cases it takes years to even get one's name on a waiting list in the first place.
So it's extremely unlikely the baby daddy landlord in this hypothetical scenario could simply make a residence available for the Section 8 program and then immediately rent it to the tenant of his choosing. He'd typically have to wait years for his preferred renter (i.e., his girlfriend) to get on a Section 8 waiting list, then more years for her to work her way to the top of the list, before he could double-dip by getting the government to subsidize his girlfriend's paying him rent to live in a house he owns.
There is a persistent misperception that the Patient Protection and Affordable Care Act (also known as PPACA, the ACA, or "Obamacare") somehow allows low-income Americans to "sign up" for free medical care and/or free health insurance. It doesn't.
Pell grants won't pay for college 100%. Trust me, I know.
The Supplemental Nutrition Assistance Program (SNAP, commonly known as "food stamps") is an often criticized assistance program aimed at combating hunger by financially struggling families. Eligibility for SNAP varies based on household size, but the maximum monthly allotment ranges from $194 for a family of one to $1,169 for a family of eight. Each additional family member after that can qualify a household for up to $146 more each month in benefits, or just under $5 per person per day. The household size posited here (a single mother with two children) wouldn't qualify for $600 per month in SNAP benefits.
Also people can't just claim disability. My legitimately disabled mother has been trying to for 5 years.
For starters, one cannot simply declare herself to be mentally ill or have back problems, then sit back and collect disability payments for the rest of her (working) life. Such claims have to be documented by medical professionals as genuinely disabling conditions, and even then those living with disabilities find that securing benefits isn't quite so easy. Presuming the woman in this anecdote is fairly young, she would have to have accrued sufficient work credits to qualify for lifelong disability payments (an unlikely scenario), and then she would still have to meet separate criteria for a qualifying condition (which precludes having earned more than $1,090 per month in that year).
Also, the "perfectly legal" part is not legal at all. Falsifying documents, address, faking taxes are all highly illegal.