I just had to post that the first dog we ever had when I was a child was a buff colored Cocker named Angel. She was the sweetest, most easy going dog ever. My mom used to say that a stranger could walk into the house and she probably wouldn't care! She had to be put down just shy of her 11th birthday due to major hip problems, among other things. I had just gone off to college at the time, so I didn't have the chance to officially say goodbye...
She was not without her health problems though. Cockers, like many dogs with long ears, tend to get ear infections frequently. This is not an inbreeding problem, just a fault of the breed itself (like little dogs having teeth problems). Unfortunately, the popularity of some Disney movies has been a curse for some breeds and Cockers are one of them (thanks to Lady & the Tramp). The inbreeding many people speak of occurred because demand was so high...in the 80's they were the #1 breed for many years. Still, I think the claim that some people make about Cockers being bad with children isn't completely true. It's not just about getting a pure-bred Cocker, it's about getting it from a breeder who cares. Many good breeders with champion dogs helped to ruin the breed by overbreeding to meet the demand for them. My mother bred our Cocker just once, but has since moved on to our Maltese and now breeds them on occasion. She is very careful about 1.) making sure our dogs are not inbred (ask for AKC papers to cross check this) and 2.) will only sell to people after interviewing them to make sure they can handle the demands of the breed. Any good breeder will do this; just because they are charging alot means nothing.
Any dog can be mean, regardless of breed. This can happen because of the way the breeder treats the dogs, or even the way you let others treat your dog (letting kids pinch, pull and tease a dog is NEVER good). There are certain breeds which are not good with children (papillions are one I know of) but Cockers are certainly not one of these! Ours got along great with our Maltese, so I don't think you have to worry about your Chihuahua either. Just do your homework, find a breeder you are comfortable with, and make sure you get a chance for both you and your family to interact with the dog before you settle on buying it.