How, exactly, is that (bolded above) possible since a far larger proportion of illegal immigrants lack an HS education or speak English?
Because there are a lot of very low paying jobs that really don't require either of those skills that nobody in this country really wants to do.
If we expect illegal immigrants to support themselves (and their dependents?) without public assistance then why not expect (demand?) the same of US citizens?
For a bunch of reasons...
First, because the immigrants who can't find work and can't survive here go home. We saw during the great recession that net migration actually went negative for a while. The lack of jobs sent millions of immigrants back home. When you're a native citizen however and you're struggling it's not so easy to just pack up and leave, not to mention the fact that you shouldn't feel like you have to.
Second, nobody is saying that every immigrant pays for themselves. Simply that the vast majority do, and their taxes cover whatever expenditures that remainder need. Just like with the native population about 95% of people looking for work can find it most of the time, but there will always be a percentage of people that are struggling to find work and in between jobs. Helping these people so they don't fall into complete destitution is a smart thing to do because it increases the odds that they'll be able to find work down the road.
Third, almost no immigrants coming to the country are handicapped, disabled or elderly. Contrary to popular belief among the right it isn't murders and rapists coming here. It's relatively young, healthy, hardworking and entrepreneurial people. They have to be in order to make the journey. They're going to have a much easier time finding work that can pay for themselves than many natives with health issues.
Lastly, you also don't have a lot of single mothers coming here by themselves. When women come with children it's almost exclusively with a husband and other family members as well. That makes it possible for them to work without needing things like daycare.