1 - the cheapest way for family of 4 to get from Seattle airport to port?
Take a taxi. It will be cheaper than anything else.
2 - will the 4 of us (2 children, age 6 and 10) be miserable in our room? 2 flip-down bunks....suites were gone when we booked.
You guys will have the time of your lives.
3 - can my 10 year old join my 6 year old in the 6-9 age group kids club? I don't think they'll want to be separated...
I suspect that they will be in different groups, but you may be able to pull it off if it is a real issue.
Have a great cruise.