Suppose you sue a company, and they came back just before the trial is to start and say, “We’ll offer you $X”. You say, “Sure”. Can the company then appeal that settlement, to an appeals court and say “The dollar amount is too large” or is it absolutely 100% binding.
I don’t think it’s considered an appeal, but you can sue again if there was misrepresentation or failure to follow through with the terms of the agreement. I’m not clear on whether judges have to approve settlement terms but they do have to accept that a settlement ends a case.
ETA: I know in divorces and child custody cases judges do approve settlements. Other than that I don’t know if a judge would have a say in the dollar amount of settlement although I’m sure there are legal requirements of a settlement agreement the judge will require before acceptance.
No, you cannot appeal the terms of the settlement. It’s a contract. Sometimes there are disputes between the time of the agreement (yesterday, for Dominion v. Fox) and the final paperwork being drafted and signed (14 to 30 days typically) about certain terms. Those are usually worked out. Sometimes there is litigation about that*. . But if there is a final settlement, it’s a valid contract and there is nothing to appeal.
*I had to file a motion to enforce a settlement when the defense tried to add a term (confidentiality) in the final paperwork that was not in our previous agreement. They argued it was a “standard” language. The judge disagreed, entered a judgment against them, and made them pay our fees for having to bring the motion.
IANAL, I assume that divorce/custody is different because it’s more than an agreement between parties, it requires (or, should have?) a court judgement stating the terms. I suppose in an amicable divorce, the terms would still be part of the court decree of the divorce? I assume in an amicable custody arrangement, you still need a court decree(or do you?) stating who has custody for other legal reasons like school and medical issue? (In some places, for example, the split of communal property is requied to be cerified by the court?)
Whereas my understanding of a lawsuit like Fox v Dominion is that having agreed to something outside the courts, they sign a contract and tell the court “thanks for being there, but this no longer should concern you”…
If they decide to dispute further, then they go back to the court, but the contract becomes part of the evidence. It’s kind of difficult under contract law to say you signed a deal in good conscience and full knowledge of the terms, but now have changed your mind. You need a better reason.
Settlements that require court approval include class actions and minor settlements. I’m not sure about family law, and it probably varies. But anything that involves children would very likely need court approval.
Usually a settlement includes a “release and discharge” which says the settlement cancels all claims related to the dispute.
Note that in this case Dominion has settled only with Fox. They retain the right to sue individuals including the My Pillow buy and (I think) certain Fox newscasters. And the other voting machine company is still suing Fox. I imagine they can even use this settlement as evidence in their case. This has not ended.
I would be very surprised if the settlement didn’t include a release for all Fox employees and agents. They can still go after people like pillow guy, and Rudy, though.
Unlikely.
Thanks for the answers everyone. 
They are:
Dominion Voting Systems still has pending lawsuits against right-wing networks Newsmax and OAN, as well as against Trump allies Rudy Giuliani, Sidney Powell and Mike Lindell. - SOURCE