I usually travel in green cars on the shinkansen - whenever I do, they are always far emptier than the regular cars, so I don't think sitting together will be much of a problem. You can always just choose a train with seats available together - you're not forced to take one train or another and the ticket counter clerk will tell you what's available.
Japanese kids, while as unpredictable as anywhere else, tend to be generally better behaved than western kids in my experience, even from a young age. And if they do start to cause a ruckus, one parent will generally remove the child from the situation until he/she calms down. So I think that would be the expectation (especially in a green car), whereas in the west both many parents and a lot of passengers would have a "oh, they're kids, what are you gonna do?" attitude. If you're actually conscientious about it, I don't think it would be a problem. But it might be a lot of "work" for you.
The luggage might be more of a problem. I agree with others about sending some bags ahead. There's only really luggage storage behind the seats at the end of the car, and there's really only storage for four bags there for the entire car. You might get lucky and nobody else will have big bags (often Japanese travel light on this route), but it's a crap shoot. I'd suggest sending the big bags ahead and keeping some airliner carry-on size roller bags with you that will fit in the overhead rack.