Bichon Frise - known as a "non-shedder", you wont see clumps of fur in your house from them - BUT they DO need almost daily grooming with a metal comb to keep their coats tangle free. Because they are a "non-shedder" it means that their coat wont flake out like the "natural coated breeds" BUT it does fall out but will stay IN the coat and bundle up in there. Hense why you need to groom them with a metal comb or a slicker almost daily.
As far as health goes with them. I used to groom and most of the older Bichons that came in usually grew warts/moles/lumps in places. They are prone to eyes that tear - so keeping the eye area clean will help keep the stains away. They also seem to catarac in their eyes as they get older.
Border Terrier - These guys are fairly hardy and being a terrier dont show pain that much lol They also dont shed all that much, should be taken to a groomers every 6 months or so to keep them looking nice though..
Dachshund- From what I have seen from things at work (vet clinic) they are prone to back problems - get obese very easily so you have to watch their food intact. Staying/going obese also puts them at risk for diabetes but other then that seem fairly healthy. I'm honestly not sure about their shedding..They dont seem to shed all that much, more so then the first few though.
Labrador Retriever- They can be prone to skin allergies/flakey skin. MAKE SURE the parents were checked for OFA hips and elbows. As well as their eyes checked as eye problems can be common in this breed as well.
Poodle(toy) - Again, get this breed from a well known breeder. They are prone to the watery eyes - so you have to keep their eyes clean or they will get stains, they are prone to skin problems, heart problems and can have sensative stomachs.
thats as much as I have seen through work from these breeds though.
now, although ALL these problems aren't common.. that is what i have seen a few come in with. If you guy your dog through a reputable breeder you really shouldn't have too much trouble as long as ALL health testing required for each breed is done.