I got three thousand dollars from my mothers life insurance
, does that mean her debts are paid?

Assuming you got the money directly from the insurance company - which I am sure you did - it means that you were the beneficiary on a life insurance policy. Her debts will be paid by her estate - the life insurance is NOT part of the estate.

You don't unless you are over the estate. Agreed, if there was a life insurance policy and you were the beneficiary it just comes to you, not used to pay any debts.
And it is not your duty to pay her debts. Money in or going into the estate will be used for that