What if it takes more than 1 minute to do the set? For example if I spend 1 min 10 secs to do the set, do I rest 50 secs afterwards before starting the 2nd round?
No, you absolutely have to choose exercises that take less than a minute, otherwise the EMOM makes no sense.
The purpose of the EMOM is to stay within fixed parameters, varying intensity, number of repetitions, weight if you use any and recovery times.