just some info. to clarify-
public assistance (as in cash/tanf) does have a lifetime limit for adults-when they hit the end of their 'clock' which keeps track of every time they are/have been on aide they no longer receive assistance. children are exempt from this. there has not been a financial incentive to have additional children while on aide for several years-with only a couple of exceptions (rape or the failure of certain types of medicaly monitored birth control) additional children do not result in an increased grant.
food stamps operate totaly differently-they are not based on familial relationships. strictly income/household size-and they no way come near to providing for an additonal 'mouth' what it costs to feed it (maybe the fs go up $50 per month, but not much more-and less if the household is larger). since rent and household expenses are used in the budgeting process, if a person receives subsidized or free housing their food stamps are greatly reduced.
wic eligibility is based on income, age of the child-and medical necessity. no longer is just meeting the income criteria going to make someone eligible-they have to have a doctor who has said the nutrition supplementation is medicaly necessary.
immigrants-if they are legal citizens, then yes they are eligible to public assistance and food stamps. if they are illegal they are not-only their u.s. born minor children would be aided. and to be honest, the 'undoc's' (welfare term) that my unit handled were probably the most honest and forthcoming about their income. they reported it, they were if an employer was not paying them illegaly under the table paid taxes on it-and often had the fastest rate of getting off aide by virtue of pooling resources and moving up the economic ladder.
'welfare to work' requires that an adult be in an approved and overseen work placement or VERY LIMITED job training program for a flat number of hours per week (used to be 30 but it was edging up when i retired). if the adult fails to do so then their needs are removed from the cash and food stamp grants (so both go down). the county i worked for if paying jobs were not available had recipeints working picking up trash or at the county recycling center.
in the years i administered welfare i saw that more often than not it was'nt an economic choice on a woman's part to become pregnant-it came down to (more often than not) either being remiss about taking birth control seriously or MUCH MORE OFTEN viewing a pregnancy as a means to 'eternaly' link her to the current man in her life. there was some kind of screwy desire on these woman's (and teens) parts to 'prove' themselves to be women by virtue of 'giving' their man a baby-and to be able to refer to a man as 'my babydaddy' (i heard this term years before it was ever said on tv or in the media

).