Norway is great for families and really friendly to children. Bars are very happy about children being around, and restaurants will lean over backwards to welcome you. Often they will have a children's menu but they are usually very happy for you to bring food for them is (like ours) your child is fussy.
There are not many playgrounds in Oslo, but there is plenty of green space and for a day out going to the lakes at Sognsvann (catch the No. 5 underground/T-bane till the end of the line) is fantastic.
The roads are very quiet in Norway compared to other big countries, and the pedestrians have right of way so the drivers tend to be far more cautious and friendly when you are trying to cross the road.