I think there I something that a number of posters don't realise. If I recall Annunziata is of Italian descent and the family dynamics are different. I'm not saying this is right, I'm just aware if this as DH is half Spanish/half Italian.
I think it is great that you are giving consideration to his request to mind DN, if I were thinking about the discussion you are going to have with DB I would do my homework and have supporting documents (ok, I'm an accountant, we like "proof")
I would be armed with adverts for childminders locally with costs (when I was considering a CM for DS 8 years ago it was £50 a day where we live), I would also have some print outs of adverts for double buggies for him (bay, gum tree, local paper, local charity shops). I would offer to make the arrangements (ie running around) once he gives you the money.
Personally, depending on your financial position I would not be as worried about the cost of food and nappies (but I can afford them, you may not be able to) but I would ask for a "float" to cover food while out or classes/clubs should you choose to attend them. This was my arrangement when I had a nanny. She always kept he receipts, but I trusted her so I didn't feel the need to check them. If I knew she was planning something special I would ask if she needed extra to cover it.
Good luck. (I say this as someone whose DM has DC3 one and a half as each week and whose MIL has DC3 half a day a week. It works beautifully for us, the grannies help out if DC1&2 are sick. We would not have been able to afford DC3 without our parents help, we are really grateful to them)