Just wanted to answer even though I can't help much since my circumstances were quite different and quite a while ago. I spent 3 months in Addis about 12 years ago, pre-kids so i guess a lot has changed and also very different to live there with a family.
So for what it's worth, these are some of my thoughts.
I do know is that there is a large expat community in Addis because of the UN-ECA and the AU which will make life a bit easier for you.
Addis isn't as dangerous as other sub-Saharan African cities like Nairobi or Dar although you still have to be careful. I took public transport (the shared taxi/minibuses, not the big public buses) during the day and taxis at night when I was there and didn't feel threatened or unsafe other than a couple of isolated incidents (although I was hassled a lot as a single Western female but more in an annoying rather than an overtly threatening way).
What did get very tiring though was just the total lack of anonymity (everyone knew who I was) and being followed around constantly by kids who seemed to think I was put there to entertain them . People didn't mean to be rude, they were just very interested in me and what I was up to!
I think as a family settling down for a few years you'll find things a bit easier than I did as a young solo woman staying for a few months. One of my colleagues was there with two yougn children and I remember she was very happy with the lifestyle.
Addis has a very rich cultural scene, fabulous restaurants (Ethiopian food is really good and one positive thing about the Italian occupation is that they left a lot of their cuisine as well), great nightclubs (you'll have to learn the shoulder dance!!) and Ethiopia is a fascinating coutnry to travel around in. although at least when I was there the lack of tourist infrastructure made it challengng at times - it's nowhere near as well developed as Kenya or Tanzania.
When I was there, there were electricity cut outs and water cut outs quite frequently. Internet was also very very slow with a govt monopoly. Things could be different now.
Good luck with your choice! I'd love to go back there and visit. Lovely place with lovely people.
Anyone living/lived there with a family? DH has a possibility of a move to work at a hospital there for several years. I would also look at working at least part time in health/NGO. Children 12,10,5 - the older two remember the couple of years we spent living in Africa when they were much younger. What's it like? DH has been and I am hoping to visit next month for a look-see.