Hi OP, I'm a dog Walker and also volunteer with the dogs trust rehoming section.
it sounds like you have already reconsidered the border collie which I think is sensible. They are fab dogs but have busy brains and are very clever. They are often known as 'the dog that needs a job to do' and don't cope well with boredom. Compared to other dogs they can be less tolerant of children- no dog should have to put up with difficult behaviour from children but you do need to bear this in mind.
You mentioned a vizsla up thread, they are stunning dogs but need a lot of stimulation in addition to a lot of exercise and can become quite unhappy and destructive if this is not supplied in abundance. They tend to be more highly strung than some other dogs.
You also asked about greyhounds, lurchers etc. with regard to cats. These dogs are sight hounds, this means they hunt by sight and speed rather than primarily scent- so they have an inbuilt trait of wanting to chase and kill small furry things that move quickly. You will hear of examples of these dogs living happily with cats, but in my opinion it is a risk not worth taking. Sadly we do hear about dogs like these who have existed with family cats but have found themselves triggered by sudden movement and instinct has kicked in whereby they chase and kill the cat. They can't help it, it is in their DNA.
Also, Whilst it is true that these dogs love cuddles and chilling all day, in terms of walking their needs can be quite complex – often they need to be kept muzzled and on lead because of their propensity to chase things at great speed.
My suggestion to you would be a golden retriever or a Labrador. Many people feel working lines are less suitable than show lines, I actually prefer the working lines they are smaller and finer and their joints fair better with age. I have found both types very easy to train compared to other dogs. They tend to be good-natured, easy to train, enjoyable to walk and an all round good family dog.
Finally, I hope you don't mind me saying that I noticed you mention running potentially with your dog. This is something that it is worth being careful about. Human type running i.e. running without stopping on hard ground can be quite harmful to dogs. They will love the exercise and it will tire them out but it is quite hard on their joints and it is not a natural way for them to travel. Left to their own devices they tend to stop quite frequently to sniff and explore rather than running for long distances. I hope that makes sense.
Good luck with your search for a dog, they are the most wonderful addition our lives.
N.B.I apologise to anyone who owns any of the breeds I have mentioned and disagrees with myThoughts. They are based on trends that are seen within the dog world, plus traits that are working each type of different due to genetics. I accept that there will always be exceptions to these trends. However, I feel it is sensible to bear them in mind when making the important choice about owning a dog.