No, it's a split system. You have to have health insurance, but there's a difference between public health insurance and private health insurance. Employers usually put you with a public health insurance company, the biggest of those seem to be TK (Techniker Krankenkasse) and AOK (no idea!) but I think there are a couple more. You pay some contributions through your wages and the company pays more and that basically covers everything. Dental, medical, emergency, etc. You get your health insurance card and present it when you need something. You usually get a rep as well for the health insurance and you can find an English speaking one and ask them any questions - definitely this is worth doing. With private health insurance you pay upfront, send the bill to your insurance company and they pay you back. If you earn less than €450 per month, then you can be insured with your husband through his company. If you earn more, then you'll need your own insurance which you'd get through your employer. Being self employed is a total pain and more complex.
I'd be inclined to put them into German schools, TBH, but that's me personally. The beginning of secondary school isn't as important as the last 2-3 years, and it would be a brilliant advantage to them to be semi-or-fully fluent in German by the time they go back. It's something I didn't really pick up until moving here but being bilingual really opens a lot of doors career wise. Most people educated to degree level (ie, teachers! :)) speak enough English to be able to communicate anything important to them, and they will pick it up as they go. If you go with German schools, and your children are over 10 or 11, you'd want to look into the options as they still have a 3-tier system. Gymnasium is the equivalent to the British grammar schools, then there is Realschule, close to a UK state secondary school without sixth form, Mittelschule (less academic, finish at 14/15) or Gesamtschule, which is comprehensive ie it offers all three streams of education. I would say if you're not intending to stay there until they're 18, then it doesn't matter what kind of school they go to, and a Gymnasium might be too demanding in terms of language required.
School in normal German schools is only until lunchtime, there is the option of an all-day school (I can't remember the word now) or a Hort, which is after-school care, which you pay for, but offers help with homework etc. I don't know how much there is in the way of Hort at secondary school level.
Housing. If you are renting you need a lot of money upfront, if the company is not finding you a house. Almost every agent charges commission or "Provisiens" which can be up to 3 times the rent, and that's on top of a deposit. You also want to start looking as far in advance as possible. www.immobilienscout24.com is the equivalent of Rightmove. Look for places which are "provisiensfrei" but generally, expect to pay it and budget for it. You will see "hot" and "cold" rent - "hot" means that it includes "nebenkosten", sometimes abbreviated to NK which is various utilities. Often in an apartment, your electricity, heating, internet etc will be shared with your neighbours and so you pay a proportion rather than being with a company of your own. It will also vary between apartments - some places having a shared electricity cost but not internet, shared water but not gas, etc etc. But basically this is not negotiable and presented as a total price. If you have any help with your company over this, take it, because housing is the most annoying and confusing thing. They can also refuse you, for no reason at all. You want to start looking for houses at LEAST four months in advance, and if you haven't found something with a month or so to go, look for short term accommodation, possibly through the company, or even just a holiday let.
Because landlords can be a total pain, it's a good idea to get legal insurance. You don't need this but a lot of people have it, and if you have an issue, it will save you money in the long run. The other type of insurance most people have is personal liability insurance. This isn't legally required, but practically it is extremely necessary due to the way that the law works here. If you accidentally damage something in any way, you can be sued, this insurance protects you from that and will also cover your DC. I have ours through my bank, but there are various providers. It costs us about €9 a month.