We live in a very children-friendly (few streets and cars) neighbourhood, approximately 15 min by tube or 30 min by bike, to Stockholm city center. Within walking distance there is a large nature reserve with lots of walking and bike tracks, several lakes, with blueberries, mushrooms and apples to pick depending on season.
We have several bikes for adults to borrow, and a couple of bikes for children under 7 years old.
We have a cat, so you shouldn't be allergic. We can arrange for her to be away if you are not too fond of animals :-)
She is very sweet though. We work as artists and the flat has character. The flat is on the 2nd story in the building. The tube station is situated 200 meters from the flat. At the same distance there are grocery stores, a few restaurants, library and cafés.