It’s also worth considering one that can be attached to your pram system. We found that really useful if babe was asleep in the car. I would highly recommend the Cybex car seats. Ours have been fantastic. We started with the cloud, and then moved to the sirona. In hindsight the Sirona would’ve been perfect throughout but the cloud came in a bundle. Sirona is suitable form newborn-4 x
You can just carry baby out the hospital or put in the pram. An infant carrier car seat is generally better as will give a better fit when they’re newborn. If you have a small baby ones last multiple years can just be too big at first and therefore unsafe. And no, just because it’s isofix doesn’t mean it fits your car. Most manufacturers will have a guide re what cars it fits into. But even then whilst it technically fits, is it comfortable? My maxi cosi seat technically fit our Focus, but the passenger seat had to be so far forward to accommodate my knees would be touching the dash. And I’m not exactly tall!
You won’t get a good fit for a newborn in a multi stage seat until they’re 4-6 months of age so you’re better off buying an infant carrier and then potentially buying one that lasts longer when they’re bigger In regards to leaving the hospital, you are not obligated to bring your car seat into the hospital, you could use a pram or a baby sling.