OP I love dogs, and I'm pleased to hear that you are thinking of bringing a dog into your home so that your children can grow up with one. There are pros and cons to puppies, older dogs, rescues, pure breeds and crosses, and large or small which all need to be considered.
I've had pure breed, crosses and rescues, toy, small and large sized dogs through my life and my children grew up with medium large dogs, so I can only say what I know about.
If you don't have alot of time an older dog will need less exercise as the above pp said and potentially won't be as bouncy and much calmer. But you do need to take in account they might not appreciate toddlers or young children who are on the go all the time or who won't leave them alone.
Also irrespective of age or type of dog, will you ensure that your children learn from the offset what the boundaries are with dogs, i.e. not taking toys or food away, leaving them alone on their bed, learning the signs when a dog wants to be left alone? Sorry to mention it as I'm sure you are intending to put boundaries in place, but I think its really important for the dog's quality of life and safety of children that they do know this. The amount of times I see mothers let their children do anything they want to their pets does often worry me. My two DC were 1.5 yr and 3 yr when we got our first family dog and I always made sure the boundaries were enforced by me.
A whippet is small medium, short coat and like a greyhound needs a couple of shortish walks and then coach potato. They do moult but its short haired and alot easier for cleaning up purposes. I don't know what they are like temperment wise, but greyhounds in my experience seem to be really laid back and calm (unless chasing after small prey). Recall????
One of our dogs is a staffy cross greyhound - so broad but on stilts. Hes brilliant and I'd always recommend staffy or staffy cross. Pure breed Staffs are really quite small dogs, and they are very friendly, short coated - but would might be abit bouncy for small young children??(check out the breed). They are terriers so they do need reasonable exercise.
I'll probably be howled down but in my experience sometimes smaller breeds can be more inclined to be snappy and less tolerant and if spaniel or terrier types they are going to be probably high energy. I do think it really also depends on how they are placed in your family hierarchy and whether the children learn about treating the dog with respect and leaving it when its had enough.
The best dog I've ever had with young children is the rescue black Lab Shepherd Cross we got when my DC were little (and we still have her - 17 years old). We got her when she was around 2 or 3 years old, so abit bouncy still but not in puppy or full on adolescent phase.This type of cross has the best of both breeds and is really calm, tolerant, good natured, intelligent and trainable. In all the years we've had her and my DC were growing up I've never had any problems with her.
Temperament fantastic, very trainable and I would trust her 100 % with kids, dogs, cats, chickens etc etc. She's always had a brilliant temperament and I couldn't have had a better dog to bring my kids up with and to train them how to be dog savvy.
The down side with her that you might find difficult, is that when she was younger she needed reasonable exercise (more than what you are talking about) - but for me that meant that I just taught my DC the joy of walks in the fields around us and I got the chance to get fitter. I did alot of trekking with buggies when my two were little and they learned to love walking. She's what I think of as a medium large (so GSD size). The other problem is that she is double coated so we are a house of fluff, but regular grooming and brushing does help with that.
The other thing I would say that sometimes bigger dogs are alot more tolerant and calmer in nature than small dogs. Socialisation is obviously very important, but alot of the time that big furry giant is a very tolerant and gentle one. I love all dogs, but although I grew up with smaller dogs, having had my Lab Shepherd Cross I probably wouldn't purposely get a small or medium one again. And I would recommend that sort of cross without hesitation.
Good luck OP and hope you find the right dog for you!