1. is it 1 bedroom? usually 1 bedroom only can allow 2 person living there. If it's 2 bedroom, you can ask manager to add baby and nanny on the contract. Usually what landlord/leasing manager do is to only put the people (adult) who applied the house on the contract. But i assume it is negociatable to add your baby and nanny. Maybe they will ask you to pay application fee for your nanny and your nanny's info for filing.
2. What kind flooring are in the apt? cemet? carpet is the most common ones which should be provided. it's not reasonable they ask you to put carpet in the apt, unless you are asking them to replace the carpet? which they don't have to.
3. it's reasonable to ask you not to decorate the place with nails, and change paint color, etc. You should keep the apt same as when you moved in.
4. you can't store washer and dryer? If the inside of the apt doen'st have hookups, you can't use washer and dryer. If you just STORE them there, i am sure no one will know and you don't have ask permission. Not allowing waterbed is reasonable because waterbed might leak and cause problem.