Am I required by law to pay back money that was sent to me from my insurance company by mistake?

Yes, you have to pay back money you received by mistake, if you are not otherwise entitled to it. The fact that you were sent money in error does not give you a right to keep it. If this seems unfair, consider the following: suppose you accidentally sent your mortage payment to the wrong bank or lender--you'd want and expect them to return it to you, and not be able to keep it because you wrote the wrong name on the check, correct? The same principal applies here--sending someone money in error does not let that person keep the money.

Technically, you need to pay it back all at once--again, you're not entitled to it. Most insurers will be reasonable, though, if you come up with a reasonable plan and explain why you can't pay it all at once. They have to agree though, so do the "homework" to make the proposed plan good and your explanation for why you need time reasonable and sympathetic.

It can affect you credit down the line--right now, while you have to repay the money, there is no judgment against you. If you don't repay it and they sue you, there will be a judgment against you, and that judgment will be reported, especially if you then don't payy the judgment.