Children make it a difficult call to make. If your children are content riding in the back of trucks, waiting in line for bus tickets, etc. Public transportation will do. Otherwise, renting a car seems like a good idea.
I have traveled by train a couple of times, I would do it again.
I used to stretch METV's for 9 months. Do a border bounce a day or two before the visa expires, that gives 60 days + you can extend 30 days.
When you cross immigration land/air they should give you 60 days.. But you want to double check they actually give 60 days. Those guys have to deal with a lot of people and mistakes are easy to make.
'onward travel' - they don't care how you leave, only that you have a plan to leave.
Allow a little extra time checking into your flight. If they question you airline or immigration you can go to back of the line, buy a bus ticket anywhere, get back in line.
I generally skip the headache and buy a bus ticket in advance.
You could get by on $1,000 US a month, but you may not like it.
$500 month rent gets you an small ok place in the city. AC $100 month, $300/month food ($10/day - good enough to survive). Unlimited internet $50/month.
You can do it for less than all that but I doubt you will want to.
Apparently, some poor sons of bitches, of low tastes find rent at a fraction of the cost. I guess you could live under a bridge for just a few cents a day.... But do you want to?