There IS a genetic link to poverty, but not to anywhere near all cases of poverty. You can take a young couple with very high IQs and excellent health (i.e. very good genes), set them to having lots of babies starting in their teens, before they have any education or any way to earn a decent living, and chances are very high that not only they, but their grown children, will live in poverty. Take the same couple, have them wait 10-15 years, finish their educations, get some good work experience, build up some savings, and then have 2-3 children, and there is almost no chance that either the couple or their grown children will spend their lives in poverty. Even if they have a baby or two in their teens, their chances for escape from poverty are much better if they don't keep on having babies -- since most of these kids are lacking education, common sense, and impluse control, they are most likely to accomplish the no-more-babies goal if they have themselves sterilized -- otherwise they will continue to have one "accident" after another.
There is a genetic component to poverty in many cases. For example, there are several impoverished rural areas of the U.S. where a huge portion of the people are genetically predisposed to low intelligence and poor health, and they tend to reproduce a lot because of a combination of welfare keeping them from actually starving to death, and the lack of ability or inclination to do anything more challenging than having sex. Same with inner-city public housing projects. Most people with any brains or physical energy up and left these hell-holes a long time ago.
But since poverty is often (perhaps usually) not directly related to genetic endowment, support for voluntary/incentivized sterilzation programs does not imply a conclusion that poverty is always a genetic disease.
Depends on if the woman is having kids or not. We're learning that on the female side fertility is like any other physical activity: practice makes perfect. Women that don't have kids lose the ability very early, women that keep having kids keep the ability for a long time (my grandmother's sister had her last of 13 at age 46, no complications).
Got any back-up for that assertion? As a 42 year old who recently underwent several cycles of IVF/embryo freezing for use down the road (entirely elective; no attempts to become pregnant the natural way, at any age), I am extremely well-read for an amateur on the subject of age and female reproductive abilities. I've not seen a single word anywhere to suggest that any reproductive medicine specialist suspects a connection between fertility in later years and the age of first and subsequent pregnancies. I could be wrong, but unless you can point to some actual research backing this notion, it must be assumed to be an old wives' tale that you picked up somewhere.
Another thing to consider when deciding who should have the nip and tuck based on the possibility of wanting to reverse that decision is that tube tying is easier to undo than vascectomies.
Alternatively -- more reliably, though more expensively -- IVF eliminates the need to reverse the tube tying, as the tubes are completely bypassed. There are also techniques to extract sperm from men for in utero insemination, without reversing the vasectomy.