I also find myself wondering why, but that's probably futile. I was discussing this with a friend at work, who's fluent in both Japanese and Chinese, and she says that Chinese has even more counters, if it's any comfort. She was also pointing out that there are times in English when we use counters, too--you wouldn't say "I have two corns", you'd say "I have two ears of corn".

You do always include the counter word. My textbook has some usage samples that include "How many cats are there?" "There are two small four-legged animals." and "How many pencils are in that box?" "There are six long cylindrical objects."