Hello, one of friend had baby boy here, so i was helping to him with this. so i know more less what i am speaking about
here after i call "You" for one of parents
01. before the labor you have to give the name of the baby to hospital and they register in hospital records, then they will send to city office
02. after the birth YOU go to city office and ask the birth certificate , all you need to know is birthday of the baby and the name of the baby and you have to carry all the identification papers. you and your partner(** important )
03. they will prepare the birth certificate for your baby, that time and if you need additional copies you pay there (if i remember correctly 1 euro each),
Additional: before you leave the hospital with the baby its important to find the doctor for your baby , because hospital need to register that doctor as a your baby's doctor for future checkups.
that is all, wait one more thing !!!!!
more importantly, you can go to another division of city office and ask the parental allowance , monthly you get 220 EUR (something around that money) + 700 EUR (only once) , monthly 220 EURO you get until your baby 3 years, after 3 years the allowance will be reduced to 87 EURO
** again the amounts and terms and conditions are subject to change , in Slovakia they change the law every year, so you may need discuss this with city office
Strongly , if you are really lucky you will find English speaking person in slovak city office , so better go with native.
Good luck for all !!!!