You situation sounds pretty good to me, so long as you are sure you could fit the dog's exercise needs around the children and their activities even when it's bloody freezing and the wind is blowing horizontal sleet along the streets. Dogs do cramp your style: you can't suddenly decide to stay overnight if you've left Fido at home, and if you plan to be out all morning, you need to factor in the dog walk.
Also if your DC are very young, you might want to give it a year or two, so you have the time you need to devote to training the dog: you are unlikely to be able to find a trained adult dog with no issues in need of a home, though you might be lucky.
As for leaving the dog, my personal view is that six hours once a week with a visit at lunchtime is fine (I do this to mine; the old geezer is past caring and the younger dog is one of the happiest animals I have ever known). Do think, though, about the likelihood of future changes in your working patterns. If you were to go back FT, would you be able to find a dog walker, for example?
When it comes to breed, consider how active and fit you are and what you want to do with the dog. If you want a few games in the garden and half an hour in the park, get something small and low maintenance. Do you want to spend time hoovering up after a mega shedder or going to the groomers once a month? Meet a few dogs - talk to your friends, approach strangers in the street, go to something like Discover Dogs. Be aware that in some breeds, there is a marked split between working and showing/pet strains (the working cocker and the show cocker, for example).
Once you find some breeds that suit you in terms of temperament, exercise needs, health (please consider health, as some breeds have massive issues, especially those with flat faces) and overall look, contact the breed rescues. A lot of rescues foster dogs out and have a very good idea of the temperament of and issues with the dogs as a result.
But you might decide, after all that, that a crossbred terrier or lurcher from a local rescue will suit you just fine. A lot of rescues, though, will not rehome to families with young children.
And finally, don't forget that dogs cost money. Day to day it needn't be a lot, but a broken leg or a mystery illness can cost a fortune: if you don't plan to get insurance, have a savings account for emergency ops and so on. Oh, and kennels when you go away. That mounts up bloody quickly.