I've given birth once in the Czech Republic. There are strict rules surrounding birth (time to theatre, certain equipment, etc), so homebirth is basically illegal. Some midwives do do it, but it's not common.
You're seen by your gynaecologist throughout pregnancy with monthly scans (12 and 20 week screening additionally), then transferred to your preferred hospital a couple of weeks before your due date.
I chose a relatively small hospital, which boasted about its mother-centric ethos and low caesarean section rate. I didn't get much experience of the labour department though as I gave birth 90 minutes after arriving! I did feel rather pressured into agreeing to an episiotomy though ("small cut or biiiiiig tear, hmmm?"). I didn't have to ask for delayed cord clamping. I can't remember how long they left it, but perhaps 5 minutes?
My son was born on Monday morning and we left hospital on Friday morning. Normal delivery, nothing wrong with either of us, but that's what was expected. Accommodation was rather basic for the first couple of nights (I was in a double room but on my own), but then I moved to private room (paid), which was more like a hotel room (en suite, electric bed, seating area, etc). Food was the same, but delivered to my room rather than having to go and retrieve it from the common room on the corridor!
They did various tests while we were there - hearing, and a hip ultrasound for my son, and another ultrasound for me to check my uterus was shrinking normally.
I'm due number 2 in May so we'll see if the experience is the same! I don't want to stay in hospital for a week if I don't need to though!