In most places you can contest the dangerous dog ruling, but in the end it it your responsibility to keep you dog under control and if she got out and caused damage then there may not be a lot that you can do about the ruling. It would be worth contacting the ranger/pound and asking what appeal rights you have - they will be required to tell you and generally if you seem genuine will tell you what they think your chance of success is.

Unfortunately, once a dog has a taste for things like that it is often very difficult to make them change their ways and it may be that if you fix the fence and replace the chooks you are facing the same situation again in the near future.

In relation to your neighbour and the chooks, I would have thought that it may be better to try and keep him on side - it may be that he could assist with getting the dangerous dog ruling overturned. The council will be able to tell you how many chooks he should be able to have but you can't force him to keep them in a certain area - unless there are rules in your area about how far their enclosure has to be from fences, dwellings etc - there is here - they can't be less than 8 metres from any part of a house.

Good luck - I hope that you can get it all sorted out.