My dd doesn't have the same challenges your young ones do, but did just start kinder this year. It seems like a lot of homework through the week... so I've requested it be sent home on a Friday so we can get a jump start over the weekend. Her homework is usually given on a Monday due the following Friday, so these extra few days have really helped us.
Your 10 and 7 year olds are still in intermediate/elementary and this shouldn't be too much of a burden to ask their teachers for. The 13 year old, in middle school I'm assuming? That might be a bit more of a challenge. With their personal challenges, one of the modifications the teachers should be making is giving your children more time. I'd definitely ask for more time either before or after an assignment is given. But make it a reasonable amount requested - it's hard being a teacher and grading all those assignments, and then to have to go back and grade one separately is challenging too.
A weekly or twice weekly backpack clean out would be good for organization. The amount of paper coming home from school is just crazy. Speaking as a former middle school teacher, I know it's a huge challenge to keep those kiddos organized, and at a time when they really need to be doing it for themselves. But it's your 7 yr. old giving you the biggest worry, and besides a binder with sections or folders for each subject and constantly being on top of it, I don't have many concrete solutions for you.
I hope you get it worked out. I know what it's like to feel you're always on a kid's case, when all you really want to do is enjoy them and love them.