Hi Brightdreams.
I agree with a number of posters above. Detroit/Shoreway, Ohio City & Tremont are all very cool areas that meet many of your wishes, but are neighborhoods in the city of Cleveland, and the Cleveland Metropolitan School System stinks. (yeah, they're trying to improve, but they've been trying for YEARS and it just gets worse) There are a few charter schools that are doing ok, but it's not the rule. If you move to these areas, count on sending your kids to private schools. I think it might be worth it, cuz these areas are just that awesome.
Lakewood is a city on the west side. Pretty nice, decent schools, right on the lake, meets just about all your requirements, I think. For some reason, though, it just doesn't appeal much to me, personally - can't explain it cuz I'm not sure why. Maybe too many up/down duplexes for me?
University Circle is a neighborhood in the city of Cleveland (same CMPS school issues already mentioned), it is walking distance to many excellent museums, great restaurants, and 2 of the best hospital systems/health care facilities in Ohio. The neighborhood shifts quickly in the area, though, so you gotta be careful if buying property/choosing a home. Seems like areas much north of Ashbury (close to Superior Rd/Rte 6), west of MLK Jr. Drive, or east of the train tracks, gets dicey pretty quick. There's a lot of blight & foreclosed homes just beyond the region "IN" University Circle.
Avoid East Cleveland (separate city) like your life depends on it.
Maybe check out the Collinwood neighborhood, in Cleveland city (Cleveland Metropolitan Schools), NE of University Circle. Collinwood has old houses (that are pretty cheap), close to museums/shopping, public transit, some really neat arts & music communities popping up. This area wasn't too great a few (>10) years back, but it is improving - might be well on it's way to gentrification in a couple years. But - still private schools for the kids.
The city of Euclid might be worth considering. It gets overlooked a lot. There's old homes, it's east of Cleveland, on the lake, very walkable, public transit, etc., and pretty close to University Circle & museums, music. Euclid was originally a very ethnic working-class area (Polish, eastern European) Not sure about the schools, though. (probably a little better than CMPS)
Little Italy neighborhood is just SE of University Circle - it's nice, old, but yards are TINY, and Cleve Metro School System.
Just past Little Italy is Cleveland Hts., which is a separate city & is getting a number of mentions above. Houses are cheap because there's LOTS of foreclosures there. Some sections of Cleveland Hts. are awesome, some are almost blighted. I think they're working on improving some of the not-so-good areas, so might be better in a couple years.
The school system is better than Cleve Public, but still pretty bad. Slid into academic watch a few years ago, missed (well, denied existence of) my son's pretty obvious learning disability. We lived there 6-7 years, moved away 3 years ago because of the schools & our neighborhood were going downhill - abandoned houses & my teen stepson was mugged and beaten for the $3 in his pocket while walking home, on our street. I miss our cool old house, walk-ability, proximity to lots of cool stuff to do, but we now have a yard, feel safe & kids go to a great school. (more on that in a bit)
University HTS is a separate city, east of Cleveland Hts - they share a school system, but more consistent neighborhoods.
Now, Shaker Hts (separate city) is similar to Cleveland Hts, but with better schools. There's also some very good private schools nearby. I'd recommend eastern side of Shaker Hts, or the area right by Shaker Square (which is on the edges of Cleveland city & Cleveland Hts). The RTA rides right through.
I don't know much about these areas' regulations about chickens & bees, but you could likely find out more online pretty easily. There's not much south of these areas that have old homes and walkability.
Chagrin Falls is much farther east - it has a LOT of what you're looking for (Excellent schools, very safe, cool old houses, walkable, quaint downtown area with good restaurants and shopping, public events, very strong community-vibe here) but there's no public transit (unless you drive 15 minutes to a park-n-ride), and music venues & museums are about a 20-30 minute drive. It is surrounded by lots of beautiful parks. Just outside of Chagrin Falls, there's South Russell (quite rural, farms and large yards), Bentleyville, Solon (newer suburb, good schools, houses mostly 1950+), Aurora (Great schools, a few older homes). Not much public transit in any of these areas.
We moved to Chesterland/South Russell area in Geauga County about 3 years ago. It's beautiful out here, some cool old farm houses, many folks have chickens, animals, farms (big & small), the schools are great, and it's very safe out here. But, it's just that - OUT here. I miss walking to restaurants & shopping. But - museums & the more concentrated areas of Urban life (in Cleveland) are only about 10-15 miles away.
Hope this helps. Good luck!