OMG when I saw this post, I knew I just had to reply!
My parents bought me a Lhasa Apso when I was 8 years old and unfortunately
I had to put him down this past January because of his old age. He lived to be 17 years old and he was the best dog my family has ever had!!
Of course at 8 years old I wasn't able to help in the decision of what breed of dog to buy. But now at 25, I'm looking to get a dog of my own and I'm highly considering getting a Lhasa Apso again. The only thing holding me back from going with the same breed is I don't want to compare the new dog to Coco.
Now to answer your questions:
Are they easy to train? My parents didn't have problems training him at all. But as with all dogs, you need to get on top of the potty training early and don't rely on pee pee pads. I have friends who are doing this with their dogs and they are having problems with the dogs not wanting to go outside.
Do they get along with Cats? Sorry, can't answer this one for you. We aren't cat-people so we never had a cat when my Lhasa was around. He did however get along great with other dogs, as we did have another dog in the family and he was great with the neighbor's dogs.
Are they playful? Definitely! We completely spoiled him with all kinds of toys from pet stores and its funny because some of his favorite toys were stuffed animals and a tennis ball tied in a sock! He loved his stuffed animals and he'd hold them between his paws and lick them. We called them his "babies" because he would take care of them as if they were!
Are they bad? Well depends on what you mean by this. As a puppy, he did get into things like chewing up my moms high heels but I think thats a phase most puppies go thru. He definitely wasn't a barker and he didn't jump up on people when they came over. He never snapped at us or at people who came to our house. When people came over, they would always tease (sarcastically) that he was such a good "guard dog" because he'd never bark at them, he'd just wag his tail and lick people, no matter who they were.
How big do they get? His highest weight was 20 lbs.
Are they hard to maintane (grooming)? Well they do need to go and get groomed because they are long-haired dogs. This can be costly, but its part of what comes with breeds like this. During summer months, we'd have him get shorter hair cuts to keep him cooler and to cut costs down a little more. He did get baths in the bath tub (and he was always so good) and he would get brushed every couple days. This is something you'll need to keep up with bc they have thick, non-shedding coats and sometimes this breed can get matted knots in their fur that will need to be cut out if they are too thick to be brushed out. Coco loved to be brushed (esp on his hiney and under the chin) and when we'd bring out his brush, he'd come running to our side because he liked it so much!!
Are they good with kids? He was great with me as an 8 year old and I have to admit, I was a pain when I was younger and my mom would yell at me for "teasing the dog". My parents always said they were surprised that he never snapped at me or anything like that.
Any health problems in this breed? The only problem we had with him was his underbite. Lhasa Apsos have underbites and their bottom teeth stick out ALL the time. Due to placque (sp?) build up, his teeth got a brownish build-up but that was taken care of with doggie toothpaste and brushing his teeth every so often. He wasn't exactly crazy about having his teeth brushed, so we didn't keep up with it all the time.
Do people have allergys to this breed? Actually, quite the contrary. I'm allergic to dogs and cats that shed. My parents did a TON of research on non-shedding dogs and Lhasa Apsos were one of the few breeds that would work with my allergies. I have very bad allergies (I become a sneezing, eye-watering maniac when I go to friends houses who have dogs and cats) and I had no problems with my dog.
I dont think there is any other advice I can give, other than you will love this breed!! My dog was an absolute love bug and other people who I know and have met that have Lhasas always seem to love them too!! You won't be disappointed!!
He has done pretty well with the potty training too. We take him outside and he usually goes! I can't think of anything negative about this dog. They are just so dog-gone CUTE!! If you decide on getting one, I hope he/she is as good natured as mine. 
They are extremely loyal, and mine was quite playful. I did not have a cat when I had him, but he did get along fine with other dogs, as long as they realized he was in charge. He actually treated the other dogs in a very motherly way.
