How do I make enough money to live and keep a roof over my head?
Solution 1:
First, if you accept the 1000 dollar EZIC gift, you will be found out 100% of the time... The only tactic to support your family is to get really good at examining the people quickly. For every successful entrant/reject, you get $5.
Just have to get really good at identifying the seals, required paperwork, identifying fake city names, or city names that have the wrong province, etc. It takes practice. There is a cheat sheet of the visual stuff somewhere, but I haven't played in a while, but google may turn something cool up for you.
Edit: This is the kind of thing I'm talking about. There are lots out there, so find one that works for you.
Solution 2:
You can also alternate between not feeding your family and not paying for heat. Your family will always be either hungry or cold this way, but it seems they they don't really bother, as long as either status lasts at most a day. (Maybe pay for both if someone is sick)
Also note that you won't lose any money for the first two tickets you receive. So there is no need to be too pedantic (especially with hard to check stuff like issuing city if you don't know it by heart) until you got your second warning.
There are some more regularities that can help: there is always at most one thing wrong with the paper given to you, so if you have to ask for one of the documents and then receive it, everything else will be fine; the first character (sometimes the first two) usually introduces the new rules of the day, so you can usually skip all checks that you already had to do the day before; if you get a card for a brothel, all paperwork seems to check out (as far as I have played); and probably some more patterns I have not yet noticed.