Tricky question. I absolutely don't agree the need of a 'house dog'. A dog by nature does not need to live within the house environment to thrive and it is not hygenic or practical and I have owned dogs all my life also. It is also a fact that dogs who live within the house have a shorter lifespan. They need to be exposed to natural bacterium to create a hardy constitution. They do not need to be treated as humans which many people assume. Ultimately dogs will accept whatever pattern you impose upon them. If you decide to bring them into the house on a particularly cold night and sit them in front of the fire then that is their treat, they will readapt to whatever structure you create.
I disagree with levelpar's post ie - your pet can be completely kept outside and should be, I agree though on their point that a 10 year old's interest may wane and you have to want the dog yourself as the adult and be prepared to mind it if they lose interest and you may know how likely that is from your own child- they may actually adore that pet until it dies or else it's just a fad. I think from experience the 10 year old will take on this pet and know their own mind. If it's a small garden it has to be taken out daily and you need to have a green area near you.
As for veterinary fees, you need to do all the initial innoculations if you take on a pup, but beyond that, the only forseeable expense is something that may go wrong unexpectedly. There is no other expense.
As for breeds there are two ways to take this. Obviously you need a small breed as you have a small garden. If you have a younger child than the ten year old in the house then be extra careful. No matter how loved or placid any dog appears to be - a new dog will be seeking it's place in the family hierarchy as the law of nature. It will respond to one person in the family more than any others as the boss and that will become apparent over time. Usually the person who feeds it.
If you intend to spend money on a breed (and that will be a be few hundred quid) then I would recommend a west highland terrier, equally comfortable indoors or outdoors and a very easy-natured breed. Daschunds, very easy to fall in love with but not as hardy and may have some health problems, but suitable for a child. Jack russells are one of the most adaptable small breeds, if you get the dog young then great, train it, but they need exercise. They are very ok being left alone, as they entertain themselves through a leaf falling off a tree to stalking a bird in the garden . And that yorkshire terrier thing! They are a self-sufficient breed, but are portrayed like poodles, they are a well rounded dog, corgis are ok too but can get lazy, but good with people.
Alternatively go to an animal welfare society and explain the situation. Contrary to popular opinion they don't just give dogs away, and may have a dog to suit your needs, as they will know the dogs personality.
Personally I would go for a female jack russell pup, who would be feisty enough to act as a guard dog (their brains are bigger than than size!) but trained and loved from the beginning and they are extremely loyal and intelligent. They are not house dogs and were never bred for that, they like pampering but are very independent.
Hope that helps
Panathon