Sounds like the 6-year-old doesn't need to be "rewarded" by watching TV anymore. Also remember no discipline rules stick until you've disciplined yourself enough to be able to bare down and follow through with each child.
But choose your battles wisely--the 6-year-old needs to stop hitting, the 10-year-old to stop talking back, and the 8-year-old to stop picking fights and being so emotional. Focus on those things first--you don't have to discipline for every mis-step or little flare-up right away, but any of those things must be taken care of. Just don't overwhelm yourself.
If they're coming home from school in bad moods, make that first half-hour or hour home really special. Watch a favorite TV show together, have a cool snack waiting for them, or take them someplace special. School really sucks, right? Give them some time to detox.
If you're not already regulating your 8-year-old's diet, try that, too. My sister's daughter has an allergy to red dye, and it causes behavioral problems for her, not physical ones. She's a totally different person once she's had something with red dye in it. If the problems stem from an allergy or hormonal issue, nothing you do to discipline will work, because chances are they don't even realize what they're doing is wrong until they're in the middle of a conflict.
Take some special time with your 10-year-old, perhaps one night a week let him/her stay up later than the other kids and hang out with you. Develop that relationship right now and it'll be the bright spot of your day when your other kids hit their tweens.
Above all else, take care of yourself. It says you're a sahm, so take those first few hours of the day to do your housework, and then do stuff for you. And make sure you get out of the house on your own terms, to do something you enjoy, at least once a week. Go to Starbucks and read a book, head to the mall to browse--anything that's inexpensive and all about you needs to be worked in.
Take care and we're rooting for you.