Your 7 yr old is being treated with FAR too much credit.
If she won't do her part of the bargain, then she doesn't get to play out alone or watch telly.
The only way she plays out is when she has done her homework. The only time that TV goes on is when her home work is done.
If she DOES play out, she comes back when she's supposed to, or she either has you supervising her like you would her baby brother, or she gets taken in. She can tantrum as much as she likes, it just lengthens the no TV rule.
The 4 yr old is picking up on the 7yo's behaviour, so get down to his level and say that under no circumstances does he EVER hit, kick or shout at you or he will go to be with no story, or won't be doing things he likes to do until he behaves better.
They are sensing they have you on the ropes here, but they absolutely don't. You have this, you absolutely do have this and can turn it round.
You are trying to stay calm, but kids are looking to find your limit. So show them! Shout! Raise your voice and show them.
Go hard but fair, and set clear periods of TV withdrawal/grounding and stick to it. Kids need to know when they have gone too far.
What you did with the homework was a good call, but it's homework, it is on your watch and it needs doing. As your dd can't be trusted to negotiate honestly with you, she loses the right to choose when she does it. Only when she proves she's mature enough to be trusted to make decisions and stick to them will she have the freedom again to make those decisions.
I say again, you can do this.
You don't mention a partner, do you have support?