If you want specifics you should go to your state's DHS or DSS website which lists the answers to these questions. however, they are mostly similar from state to state.
I agree that walls of text are too hard to read, however what I did read seemed interesting for me.
I have often wondered about what welfare really is like in the US.
My understanding is that it varies from state to state, so it is difficult to get a complete picture ... but anecdotally I have heard that in general, welfare support is less than in other comparable western nations ...
I would be interested to know some hard facts - eg ...
how much does a single woman with two children receive on welfare?
This depends on what type of welfare. You have to answer that question. If she has applied for SNAP benefits she would probably get around 170-180 dollars per person per month for food. If she is on TANF she will get somewhere around 450-550 a month. TANF does allow to be combined with both SNAP and you would also qualify for medicaid. if the woman is pregnant she would probably also be qualified for WIC which is part of SNAP. A good estimate would be that if she qualified she would get about 500-600 in food only benefits and the 450-550 in cash benefits per month.
Now she can only receive cash benefits for herself on TANF for 60 months, and most states have a requirement you have to look for work. If she loses her own personal benefits from TANF they will still pay the children's benefits until they are 19 or no longer living with the mother.
what other benefits does she receive?
Her and her children would probably qualify for medicaid. She might qualify for section 8 housing, though there is a waiting list on that from what i hear. There is also a good deal of likelihood they can get help paying their utility bill.
what free public health measures are available for children?
medicaid is there, but it technically is almost free. i do believe there is a nominal fee of a dollar or two for a visit, and most doctor's don't take medicaid.
how much is the average rent (or what proportion of her welfare payment would she expect to pay on rent)?
It is mostly irrelevant to the payment from what i have seen. It is her responsibility to find a place she can afford. There is an option for section 8 housing, but that has it's own problems if it is available at the time. the place where your rent becomes an issue in your benefits is Social Security Disability. It is there where you will receive a few hundred dollars more a month if you are responsible for your rent. For TANF it seems you get the benefits and are responsible for finding a way.
how much are some of the basic foods she would need to buy for her family?
You can survive in most places on SNAp funding if you are just concerned with eating. You cannot buy lobster and expensive steak, but you won't go hungry if you purchase smartly. They decrease the benefits of all systems as you have more people to care for because it is assumed you will combine things like food more efficiently.
what would she pay for public transport?
That has little to nothing to do with getting benefits. She would be expected to walk if she couldn't afford it.
what assistance is available to help her upgrade her skills and become more employable?
There are some state and federal tuition assistance and training programs. That is an entirely different screwed up system. In the end all that doesn't make much difference when many employers have started running credit checks and denying applicants who are behind in their bills, which many welfare recipients are.
can she get subsidized childcare if working or studying?
that is dependent on a lot of factors. This one you would have to look up based on specific circumstances of the mother and her location, and the availability of child care for needy families in the area.