Looking for advice on kid-friendly neighbourhoods of Berkeley

In Berkeley, you are not guaranteed to be assigned to a school that is closest to you. 

Elmwood and north Berkeley are nice, easy to bike. I think the areas near Ohlone Park,  North Berkeley Bart station, and Monterey Market meet your description. Elmwood feels the safest. I love the area near the School of the Madeline. 

All of Albany is walkable/bikable and neighbors north Berkeley. Walking to school is possible. 
 

I am not sure where you are moving from. I would be a bit nervous about sending kids to a park without an adult. I am not trying to be overly protective and I was a latchkey kid, but there are a lot of unstable people around here and daytime crimes do happen. If you are looking for an insulated area, Berkeley might not be the best area for you.

I do see kids walk to school together in Piedmont and Alameda and their parks are clean. I do feel safe to let kids roam more freely at parks in Piedmont and Alameda than in Oakland and Berkeley, except for parks in the hills. 

If you don't absolutely need to be in Berkeley, I'd look in Albany (with caveat that rentals are more limited there). Bike trail, walkable schools, and lovely parks. Right next to Berkeley. The challenge with Berkeley is that even if you live right next to a school, you may not end up enrolled there because of the structure of their assignment system (especially with three kids to place).

If you are open to outside of Berkeley, Alameda definitely fits the bill for everything you describe!  There are TONS of families with kids of all ages, wide sidewalks, tons of parks, and very flat/bike friendly. It is also generally more affordable than Berkeley.  

For bikable/walkable kid-friendly, I recommend the areas in Berkeley around Live Oak Park and Hopkins/Monterey. Also look in El Cerrito around Castro Park, Harding Park, and Cerrito Vista Park, and pretty much anywhere in Albany.

Not sure where you're coming from or what your comfort level is with, say, occasional unhoused people in the park. I think the area near North Berkeley BART/Westbrae is great, with wide sidewalks and extremely bikeable. And you're right off the Ohlone bike trail which you can take for miles and miles, lots of fun. I see kids your kids age by themselves at the parks in our neighborhood like Strawberry Creek, Ohlone, and Cedar Rose. If you're less comfortable with people occasionally sleeping/camping in the park, the hills might be a better bet, but that makes it not bikeable, obviously. I see lots of kids biking and walking themselves to school in all areas in North Berkeley (ours is too young for that still).

Come on down to south-west Berkeley! The neighborhoods between San Pablo Park and Strawberry Creek park are great. Very walkable, bike friendly, quiet streets that take you to the Greenway, many parks small and large, close to freeway onramps, and lots of convenient services like coffee shops, restaurants, library, pool, Berkeley Bowl, small Target, Mi Tierra, and much more. You can get places fast by car taking Sacramento, 6th, or San Pablo. 

Another vote here for central/south/west Berkeley--that could mean anything from the neighborhood by San Pablo Park to Westbrae. Conveniently located and still in the flat-ish areas, so it is easy to walk and bike around. The parks in the non-hill areas are safe--my 12 year old has been going to the parks near us (Strawberry Creek and Cedar-Rose parks, among others) alone and with friends, without adult supervision, since age 10. I would be suspicious of suggestions that only parks in the Berkeley Hills are safe, since some of them are more remote and see less activity during the day. I recommend checking the school district's zoning map so you have an idea of where your potential school assignments could be, for elementary-aged kids at least, since being assigned to a school in the hills when you live in the flats, or vice-versa, could be a bummer when it comes to transportation at least (there are buses available if you are more than a mile from your assigned school, I believe).