I'm in Denmark. You get around $1100 per month from the government as a university student, you then get around the same amount per child (not sure if a couple get half of that each). Still if you're two students, with a child, that's at least $3300 a month. That's not a lot of money, but there are also government loans you can get, and again, free daycare and subsidies for housing. It's not a get rich scheme, but it's also only meant to be temporary i.e. until you finish your studies.
That sound more like "Børnetilskud". Everyone gets that, regardless of being a student or not, that's just help to buy clothes and stuff like that for your child. It's paid out every quarter.
There's a bunch of stuff like that, some can be "stacked", some are mutually exclusive, some are "per child" some is per adult. Some are only available to single parents, some are only available if both parents are enrolled in an education, some are only available if you make less than a certain amount.
Yes and no, the government is trying to steer young people in the direction of engineering, nursing, doctors, teachers and trades (carpenter, bricklayer and so on), but it's not clear where the people are suppose to come from. Essentially Denmark is missing people in also every profession. There aren't enough people. My wife works in a field where unemployment is 12, not percent, but 12 people. So if you're unemployed, qualified to work in the EU and have a recognized education, applying for jobs in Denmark isn't a bad bet.
Various governments have also attempted to boost birth rates, but unsuccessfully.