Yes you can pay or have (cheap like 1k/year) insurance and get immediate service as well. There are private clinics that also work together with the public system.
Daycare and Healthcare for kids is cheap though. And you don't have to pay for good schools or move to another neighbourhood to get a good school. This far outweighs the cost of a car.
That's about right. To live in Helsinki you'd probabaly need minimum 1000€ for a shared Apartment and food... With 560€ you might be able to live somewhere and scrape by.