Kittens are great and really tough usually if your little ones understand they must be gentle. A young kitten can climb almost anything once it figures out how, but that doesn't mean its going to be safe for the kitten to climb. If I had really small kids, truth be told I'd adopt an older cat with a sweet disposition. The weird thing about cats isn't so much the age, is that every one of them has a different personality. Some can be really young and fearless, old and nervous, old and fearless... you get the idea. It really depends on the kitten/cat.
What you do need to do is have a LARGE litterbox. People make the mistake of getting a small one (because they don't like the smell, the inconvinenece, etc....) but you want as large of one as you possibly can, and clean it often.
Feed them good food. Cats really do a lot better on wet food thats mostly meat, not filler like corn or rice. Some of mine even get a raw food diet. I've seen a cat diagnosed with diabetes and given a short lifespan revert and get really healthy by just getting fed raw food. Kibble... like cat chow or crunchies is the worst thing you can feed them. Their GI tracks are designed for meat, not a balanced diet of ground up veggies and fillers.
Otherwise, cats are a little piece of heaven. And if they like your kids, they will be loyal friends and companions for the whole of their lives.
If it were me, and I had your range of kids, I'd go adopt a cat from the shelter or out of the paper that really needs a good home but that is older and house broken. By older I mean, I'd pick the 8 month old (even if its nervous.. that can change) or one thats a few years old. and really laid back. I have a house full (literally 14) of geriatric cats that are all over the age of ten... that are GREAT babysitters and really good around kids, other pets, etc. A lot of older cats have a terrible time finding homes and they are the ones that do the best with small kids. A small child sometimes has a hard time remembering to not pet to hard, or can easily fall carrying a small kitten and really hurt them. My oldest cat currently is 19. I had one live to be 22... so I can give you any help you'd like after you make your decision if you need any.