This is very common early on and the more time goes on the better it will be, you’ll be less tired and be able to use your free time to go out more and see friends etc. Summer is a while away and baby will be older then so you will probably feel more comfortable with dad going away for some him time, which is normal and okay to have that time! I’d compromise and book a few girlie weekends away for your you time too!
Oh hell no, I would say Yeah fucking right, you are not leaving me alone for a week while I'm with the baby. He also chose to have this baby so it is a commitment he has to stick with.