Do we have to pray as soon as the time for prayer comes?

Do we have to pray as soon as it is a valid time to do so or we have the whole time between two prayers (lets say for example its time to pray Duhr but I prayed befor Asr starts in 30 mins). Another case would be Its time to pray Fajr but i prayed once i have waked up is my salah not accepted as i should have woken up as soon as it was valid to pray Fajr? I know that Fajr And Isha because of their timing they are considered as a challenge, so me waking up later than i should is making things in my own comfort, is this haram, brothers and sisters please advice!