Hi, I live in Germany too, near Karlsruhe. We've been here just over a year.
You'll need to get a pet passport for your cat - ask your vet about it, they can sort it out for you, look into it ASAP, there are time constraints. That will prevent the need for quarantine. As to how to get them over, you'd have to look at the time/benefits of flying vs driving. Flying is obviously quicker but you wouldn't be able to be with the cat for the journey, and might have to use sedation. Driving you'd have to stop every now and again and either have a litter tray in the car or get your cat used to a harness. Either way there's obviously a risk the cat could escape especially if a well meaning child opened a door or window. If you drive I'd recommend one or two adults doing that alone while other(s) fly with the children.
Generally the child benefit is good, you also get a payment for children under one and children under three. After three childcare is heavily subsidised - your middle one is Kindergarten age (which is not compulsory but I highly recommend), and your oldest will be in the first class of school. Here, the kindergarten costs €94 per month, it's from 7.30 (up to 9.00 drop off) until 2pm (can pick up from 12.30) every weekday and there are only about four weeks of holiday in a year - one at Christmas, three in the summer, and you don't pay for the August month. I guess costs should be similar there. It's higher for under threes, and places are much harder to find, but they are around (Although reserved mostly for working mothers). This has been the best thing for DS, making friends, picking up the language. He loves it. He was 5 when we moved, he's 6 now and almost fluent.
When renting look at www.immobilienscout24.de - it's the german equivalent of rightmove. You'll need cash upfront to move in, usually commission is the bulk of the cost and can be around 2.5 months' rent - which you don't get back. It's huge. You can look for "provisiensfrei" which means without commission but they're still expensive. If the job offers any relocation package or help with finding a place, take it. Someone local to the area can probably advise you how soon you need to look in advance - in Karlsruhe you need to be looking 3-6 months before you want to move, it's crazy.