Any dog can be trained to steady around rabbits or any other animal/livestock and the same applies to being good around children. However, some dogs can be easier to train than others and some breeds or types have an (undeserved!) reputation, so if you want to childmind, I think you'd need to consider that too, as simply owning a dog would put a surprisingly large number of people off using a childminder. Also, although they can certainly still be trained to be around rabbits, some breeds, such as Huskies and Greyhounds will never really be reliable because of their high prey drives, but Huskies aren't really suitable as pets anyway, IMO, as they have quite specialised exercise requirements.
Some of the gundog breeds (cocker spaniels, Golden Retrievers and Labradors) are common as pets because they are easy to train, biddable and also don't have excessively high energy levels. They are also well liked by most people who like dogs even if they don't own a dog. I have Golden Retrievers (also a rescue staff cross) and have shown them for years and used to breed them and everyone loves them when you are walking them and wants to stop and talk to them. There are working and show types in all of these breeds and they are quite different to each other in most cases and show types are generally calmer and easier, although not always.
When we got our first dog, the breeder said 'you get out what you put in' and she couldn't have been more right. Your dog will be well behaved, won't steal food, won't chase or bother your rabbits, will walk nicely on a lead etc if you put in enough time and effort to train it to do these things. You may or may not need professional help with that, but classes are certainly worth it from a socialisation point of view.
If you intend to get a puppy of a certain breed, you can speak to several reputable breeders and visit them to get an idea of which breed is right for you. It is vitally important to go to a good breeder who has done health tests, not only for your own benefit, but also for the welfare and future of the breed and dogs overall. A good breeder who is well established with work or show their dogs (otherwise, why are they breeding them? They should only be breeding to improve the breed) and will have enough experience and knowledge to advise you if you come across any problems. They will also be willing to, in fact they will insist, that if you can't keep the dog in the future, at any point in it's life, that you return the dog to them. They will have done health tests and will KC register their dogs, but sadly KC registration alone, even if the breeder is in the accredited/assured breeder scheme, is not a sign of a good breeder, it's a minimum requirement.
You can also get a particular breed of dog, usually an older puppy or older dog, through breed rescues and may sometimes be in touch with the breeder and have their support if you do this. The obvious advantage of this is that the dog will normally be toilet trained and will probably have some basic training already.
'Ordinary' all breed rescue sometimes have pure breeds in too and puppies, much more often than you'd think, so don't rule them out. It doesn't always mean you get a dog with an unknown history and most rescues that rehome with children vet the dogs very thoroughly before saying they are suitable as such. Sadly, some dog pounds market themselves as rescues, which they are not and these aren't suitable places for a novice owner to get a dog and some rescues are not as good as others, so you do need to vet the rescue thoroughly and ask questions. If you think a rescue dog may be an option for you then I have a number of rescue contacts all over the country, so may be able to put you in touch with someone near you.
