It's actually not recommended (unless medically necessary) to start expressing until after the milk supply has been established (6 weeks). You will risk oversupply and mastitis and I would suggest to him that he actually learns about breastfeeding, rather than dictate to you about something he clearly knows nothing about. Ignore any noise from his mother.
Do NOT be bullied into overnights, it is NOT in the best interests of a baby to be separated overnight from the primary caregiver. Not only this, but prolactin, the hormone that produces breastmilk, is higher at night, and those night feeds are crucial. It's also crucial that baby feeds directly from the breast as much as possible, as their saliva sends signals to your body to produce the right milk for them. Also, by expressing rather than feeding directly, your supply will start to drop. This is why exclusive pumpers don't end up being able to do so long-term.
Unless things have changed, a judge would not normally grant overnights until 2 yrs old...but sadly, some are not putting babies best interests first anymore as more and more fathers are trying their best to stop having to pay CSA by going for 50/50.
The Association of Breastfeeding Mothers is a good organisation for advice with this.
Of course, he has a right to be involved, but a baby who is exclusively breastfed should stay with it's primary caregiver, and if he is a decent man (obviously he's not), then he will agree to spending time with baby alongside you.