I have had a dachshund, a Norwegian Elkhound, 2 german shepherds, a collie, and 4 beagles.
Hounds aren’t for everyone. They are willfull and stubborn. People who want a dog to follow orders instantly will not be pleased with them. They dig in the garden and make holes in the lawn.
But, they are affectionate and loving. They love your companionship. They will not kill an intruder but they will sure let you know if something is amiss. My neighbor loves the fact that they will make noise anytime something sound wrong or smells different.
I’ll put in a vote for Malteses. They are incredibly small and delicate but they’ve got really sweet dispositions and are very smart but also eager to please.
Note: Bostons are not classified as terriers, despite the name. They’ve never been used to chase/hunt small game.
ETA: Bad News Baboon said something similar. I concur.
I was going to recommend pugs, but I don’t think they’re small enough; they are a toy breed, but not small enough to be a purse dog by any means (maybe a tote bag dog!). I can carry my pug in one arm, but a 7 year old definitely couldn’t (even two arms would be hard). Another pug I know I can barely pick up! They are great dogs, though, and very personable. I’ve got a pug and so does my sister, and they both are happy playing with smaller dogs or much bigger dogs with pretty much equal enthusiasm. They do vary in size, but it’s hard to tell when they’re puppies; they started about the same size, but her (female) pug greatly outweighs and outsizes my (male) pug, even if body fat’s out of the consideration.
They aren’t yappy but they do seem to like to bark if they’re anxious about something if they don’t know what it is. I figure it’s little dog syndrome. I’ve tried to train this behavior out and the best I’ve gotten is volume control. The dog literally can’t seem to stop barking if he wants to bark, even knowing he shouldn’t, but if scolded/watched he’ll start making tiny woof noises with his mouth closed instead of a full-on bark.
Perhaps a Pomeranian? They can be great dogs with good temperaments, but they do need to be trained. Any dog that’s in the hands of a kid who doesn’t know what they’re doing can be a terror.
I see a lot of Poms at the dog park and have known people who kept them. They’re small, but they enjoy playing with other dogs, and don’t seem to have the anxiety and social reluctance that I see in many toy breeds. I’ve seen Pomeranians happily playing with full size Labradors; what they lack in size, they make up in plenty of personality.
Huh. A Google image seach for Maltese turns up several pics of them in the short coat Lynn mentioned. They look like white toy poodles when they’re clipped like that!
I should clarify the purse dog statement. The dog doesn’t have to be micro sized - I would say anything 15 pounds and under (prefer much under but nothing is set in stone).
To her the cuteness factor is the important part. To me, I want to be able to fit all of us in my car, a Mini.
It is my goal to take the dog to obedience classes with her. I think it would be fun.
Another vote for the shelter, or humane society choice. While you’re there, you can get a good read on your future dog’s “yappiness”.
It might also be useful to you if we mention some small breeds that we would not recommend. I would advise you to stay away from Min-Pins, at least for a 7 year old. They’re fascinating at first, but you soon realize they are almost untrainable. Obstinate doesn’t begin to describe this animal. We got one because our daughter (13) wanted something smaller to play with, but it soon turned out he wanted nothing to do with her, and she became disenchanted with him. I actually like the stubborn little cuss, and find his personality oddly charming. After a few years of effort, I’ve finally convinced him to smoke outside and to quit hitting the Border Collie with his pool cue.
As to the 3-dog dynamic, we’ve acquired another shelter dog (Chihuahua mix) and there haven’t been any major adjustment problems. They seem to have a nearly continuous 2-dog wrestling match going, but the participants change frequently (never a 2 against 1 situation). It takes a little while for them to sort out who’s dominant, but once that’s over everything’s pretty peaceful.
Best of luck.
Longhair dachshund. They’re adorable, have long brushable hair, and mine will sit patiently and let you do whatever you want to him. Hold him, and he’ll never try to get up. Never barks, and only licks just a little bit. He wrestles with my boston terrier all the time.
Maltese are incredibly cute, whether they have a long or short coat. I recommend a short coat for all Maltese unless they are actually being shown. The main drawback of a Maltese, in your situation, is that they do require regular haircuts, and daily brushing. Your daughter might or might not enjoy the daily brushing, but these little dogs are very cute and very companionable. Pugs, for instance, have great personalities, but I would not consider them cute, and I don’t think that a seven year old girl would think that they’re cute, either. German shepherds are very attractive, and very loving, but they won’t fit into your car, and they aren’t cute except as puppies.
While we’re recommending against dogs, let me recommend against Jack Russell Terriers. These little guys can be very attractive, and very smart, but you really, really have to work with them, and they’ll probably be testing the limits all the time. They have to be kept busy.
Another thing: a Maltese is quite likely to let or even enjoy your daughter dressing it up in doll clothes.
I don’t know if it’d be too big for what you’re looking for, but for years, my mother had a Sheltie (AKA miniature Collie, although they are not related to Collies in any way). He was a great dog. Laid back, easy-going, friendly, definitely not an ‘ankle biter’, and so loving that when my Mom died, we had to have her Sheltie put down because he wouldn’t stop grieving her. 
Myself, I prefer larger dogs (a Husky is my ideal dog), but if I were going to get a small dog, it would be a Sheltie.
During our first marriage and before we started supporting “spay, neuter and adopt”, D and I bred and raised Belgian Tervurens (herding dogs).
Yeah, they’re not little, but damn loyal and damn good protectors. The only problem with them is that the herding instinct in them is so STRONG, they will herd their family unless they’re obedience/leash trained and this is probably something you wouldn’t want to devote that much time to, I don’t know. Plus of course you did say small, right.
But you asked and if we could, I’d dearly love to have another Belgian. We loved our Yorick very much (lost him to heartworms we thought we’d gotten rid of).
If you’re going with a breed though, I agree with norinew (hey sweetie!:)) and get a Sheltie, but might I ask you to perhaps visit a shelter and see what can be found there? These days some beautiful dogs are killed in shelters due to their owners being vicitms of this crappy economy and having to give them up. It would be wonderful if you and your daughter could save one’s life.
Just a suggestion. 
Thanks
Q
Fear the Weener.
A Dachshund will also EAT anything, whether you consider it vaguely edible or not. I’d definitely get a vet care plan with a Doxie, at least for the first year until you find out if your particular dog is a canine vacuum or not.
I adore pugs. Pugs, as mentioned upthread, have “big dog” personalities now in a more convenient travel size. They are my favorite dogs ever. Whether or not they’re “cute” depends so much on personal taste. I always thought of them as “so ugly they’re cute”, even when I was a kid. And today’s kids, raised on a diet of Japanese inspired animation, may find them just cute cute. They make good watch dogs, which means they’ll bark (not yap, a pug’s voice is gravelly) when they see the mailman walk up, but they’re utterly useless as guard dogs, being uninterested in defending their turf. They’re all bark and no bite. Which is good for kids. And unless your pug is trained into anxiety, they don’t bark for no reason at all, unlike some of the yappy dogs. A simple, “Oh, thank you! Good dog!” and then turning your attention to the thing that made the pug bark is almost always enough to get them to stop. They know their job - alerting you, the pack leader, to danger - and they know what’s your job - dealing with the danger.
And I’ll add that “going to a shelter” and “looking for a particular breed” are not remotely exclusive. There are plenty of purebreds in shelters these days.
Believe me when I say that I believe in saving dogs. My dogs are all rescues and I vastly prefer this route over shelters. I prefer it because they get the dogs from the shelter and care for them in a home setting: they can more accurately tell you about a dog’s temperament.
I’m just saying this so this thread doesn’t turn into a “go to the shelter” thread.
The Maltese definitely seem like what she would want.
Now the question is how do you avoid the nasty fur around the eyes?
Your obedience class plan is a good one, but I’d say you’ll be lucky to find one that allows your kid to participate. Around us here (Bay area) the classes are listed as age 12+.
Don’t get a basenji!
What about a Border Collie? I have friends with two of them, and they are wonderful beasts.
The dogs, I mean, not my friends.
The OP is asking for small breeds.
I would call a border collie is a small breed - they don’t come up to my knee. They’re very bright and loyal as well. I know, they aren’t as small as a chihuahua or Italian greyhound, but they’re smaller than a beagle or a basset hound.